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^OTED    TO    THE    SCIENTIFIC  ^^r>^     AND    ENGINEERING 
»ECTS    OF    ELECTRICAL    COMMUNICATION 


U  M  E  XXXV  JANUARY    1956  tf  k  k---  • '  t.  N  U  M  B  E  R-lv 

DiflPused  Emitter  and  Base  Silicon  Transistors  J  ^'  ^   '^  ^  ^^^° 

M.  TANENBAUM  AND  D.  E.  THOMAS         1 

A  High-Frequency  Diffused  Base  Germanium  Transistor    c.  a.  lee    23 

Waveguide  Investigations  with  Millimicrosecond  Pulses 

a.  c.  beck    35 

Experiments  on  the  Regeneration  of  Binary  Microwave  Pulses 

o.  B.  delange    67 

Crossbar  Tandem  as  a  Long  Distance  Switching  System 

a.  O.  ADAM      91 

Growing  Waves  Due  to  Transverse  Velocities 

J.  R.  pierce  and  l.  r.  walker  109 

Coupled  Helices  j.  s.  cook,  r.  kompfner  and  c.  f.  quatb  127 

Statistical  Techniques  for  Reducing  the  Experiment  Time  in  Re- 
liability Studies  MILTON  sobel  179 

A  Class  of  Binary  Signaling  Alphabets  david  slepian  203 


Bell  System  Technical  Papers  Not  Published  in  This  Journal  235 

Recent  Bell  System  Monographs  242 

Contributors  to  This  Issue  244 


COPYRIGHT  1956  AMERICAN  TELEPHONE  AND  TELEGRAPH  COMPANY 


;  ,  *  -^  -^  f  -  -.r »  '  J "  -'  • 


THE  BELL  SYSTEM  TECHNICAL  JOURNAL 


ADVISORY    BOARD 

F.  E.  K  A  P  P  E  L,  President,  Western  Electric  Company 

M.  J.  KELLY,  President,  Bell  Telephone  Laboratories 

E.  J.  McNEELY,  Executive  Vice  President,  American 
Telephone  and  Telegraph  Company 

EDITORIAL    COMMITTEE 

B.  MCMILLAN,  Chairman  H.  R.  HUNTLEY 

A.  J.  BUSCH  F.   R.   LACK 

A.   C.  DICKIESON  J.   R.   PIERCE 

R.   L.  DIETZOLD  H.   V.   SCHMIDT 

K.  E.   GOULD  C.  E.  SCHOOLEY 

E.  L   GREEN  G.  N.  THAYER 


EDITORIAL    STAFF 

J.  D.  TEBO,  Editor 

M.  E.  s  T  R  I  E  B  Y,  Managing  Editor 

R.  L.  SHEPHERD,  Production  Editor 


THE"  BELL  SYSTEM  TECHNICAL  JOURNAL  is  pubUshed  six  times 
a  year  by  the  American  Telephone  and  Telegraph  Company,  195  Broadway, 
New  York  7,  N.  Y.  Cleo  F.  Craig,  President;  S.  Whitney  Landon,  Secretary; 
John  J.  Scanlon,  Treasurer.  Subscriptions  are  accepted  at  $3.00  per  year. 
Single  copies  are  75  cents  each.  The  foreign  postage  is  65  cents  per  year  or  11 
cents  per  copy.  Printed  in  U.  S.  A. 


THE    BELL  SYSTEM 

TECHNICAL  JOURNAL 

VOLUME  XXXV  JANUARY   1956  number  1 

Copyright  1956,  American  Telephone  and  Telegraph  Company 

Diffused  Emitter  and  Base  Silicon 

Transistors* 

By  M.  TANENBAUM  and  D.  E.  THOMAS 

(Manuscript  received  October  21,  1955) 

Silicon  n-p-n  transistors  have  been  made  in  which  the  base  and  emitter 
regions  were  produced  by  diffusing  impurities  from  the  vapor  phase.  Tran- 
sistors with  base  layers  3.8  X  10~  -cm  thick  have  been  made.  The  diffusion 
techniques  and  the  processes  for  making  electrical  contact  to  the  structures 
are  described. 

The  electrical  characteristics  of  a  transistor  with  a  maximum  alpha  of 
0.97  and  an  alpha-cutoff  of  120  mc/sec  are  presented.  The  manner  in  which 
some  of  the  electrical  parameters  are  determined  by  the  distribution  of  the 
doping  impurities  is  discussed.  Design  data  for  the  diffused  emitter,  dif- 
fused base  structure  is  calcidated  and  compared  with  the  rneasured  char- 
acteristics. 

INTRODUCTION 

The  necessity  of  thin  base  layers  for  high-frequency  operation  of  tran- 
sistors has  long  been  apparent.  One  of  the  most  appealing  techniques  for 
controlling  the  distribution  of  impurities  in  a  semiconductor  is  the  dif- 
fusion of  the  impurity  into  the  solid  semiconductor.  The  diffusion  co- 
efficients of  Group  III  acceptors  and  Group  V  donors  into  germanium 
and  silicon  are  sufficiently  low  at  judiciously  selected  temperatures  so 

*  A  portion  of  the  material  of  this  paper  was  presented  at  the  Semiconductor 
Device  Conference  of  the  Institute  of  Radio  Engineers,  Philadelphia,  Pa.,  June, 
1955. 


2       THE  BELL  SYSTEM  TECHNICAL  JOURNAL,  JANUARY  1956 

that  it  is  possible  to  envision  transistors  with  base  layer  thicknesses  of  a 
micron  and  frequency  response  of  several  thousand  megacycles  per 
second. 

A  major  deterent  to  the  application  of  diffusion  to  silicon  transistor 
fabrication  in  the  past  was  the  drastic  decrease  in  lifetime  which  generally 
occurs  when  silicon  is  heated  to  the  high  temperatures  required  for  dif- 
fusion. There  was  also  insufficient  knowledge  of  the  diffusion  parameters 
to  permit  the  preparation  of  structures  with  controlled  layer  thicknesses 
and  desired  dopings.  Recently  the  investigations  of  C.  S.  Fuller  and  co- 
workers have  produced  detailed  information  concerning  the  diffusion  of 
Group  III  and  Group  V  elements  in  silicon.  This  information  has  made 
possible  the  controlled  fabrication  of  transistors  with  base  layers  suffi- 
ciently thin  that  high  alphas  are  obtained  even  though  the  lifetime  has 
been  reduced  to  a  fraction  of  a  microsecond.  In  a  cooperative  program 
with  Fuller,  diffusion  structures  were  produced  which  have  permitted 
the  fabrication  of  transistors  whose  electrical  behavior  closely  approxi- 
mates the  behavior  anticipated  from  the  design.  This  paper  describes 
these  techniques  which  have  resulted  in  high  alpha  silicon  transistors 
with  alpha-cutoff  of  over  100  mc/sec. 

1.0    FABRICATION    OF   THE   TRANSISTORS 

Fuller's  work  has  shown  that  in  silicon  the  diffusion  coefficient  of  a 
Group  III  acceptor  is  usually  10  to  100  times  larger  than  that  of  the 
Group  V  donor  in  the  same  row  in  the  periodic  table  at  the  same  tem- 
peratures. These  experiments  were  performed  in  evacuated  silica  tubes 
using  the  Group  III  and  Group  V  elements  as  the  source  of  diffusant. 
Under  these  conditions  a  particular  steady  state  surface  concentration 
of  the  diffusant  is  produced  and  the  depth  of  diffusion  is  sensitive  to 
this  concentration  as  well  as  to  the  diffusion  coefficient.  The  experiments 
show  that  the  effective  steady  state  surface  concentration  of  the  donor 
impurities  produced  under  these  conditions  is  ten  to  one  hundred  times 
greater  than  that  of  the  acceptor  impurities.  Thus,  by  the  simultaneous 
diffusion  of  selected  donor  and  acceptor  impurities  into  n-type  silicon 
an  n-p-n  structure  will  result.  The  first  n-la,yer  forms  because  the  surface 
concentration  of  the  donor  is  greater  than  that  of  the  acceptor.  The 
p-laycr  is  protluced  because  the  acceptor  diffuses  faster  than  the  donor 
and  gets  ahead  of  it.  The  final  n-region  is  simply  the  original  background 
doping  of  the  n-type  silicon  sample.  It  has  been  possible  to  produce  n-p-n 
structures  by  the  simultaneous  diffusion  of  several  combinations  of 
donors  and  acceptors.  Often,  however,  the  diffusion  coefficients  and 
surface  concentrations  of  the  donors  and  acceptors  are  such  that  opti- 

1  C.  S.  Fuller,  private  communication. 


DIFFUSED    EMITTER   AND    BASE   SILICON   TRANSISTORS  3 

mum  layer  thicknesses  (see  Sections  3  and  4)  are  not  produced  by  simul- 
taneous diffusion.  In  this  case,  one  of  the  impurities  is  started  ahead  of 
the  other  in  a  prior  diffusion,  and  then  the  other  impurity  is  diffused 
in  a  second  operation. 

With  the  proper  choice  of  diffusion  temperatures  and  times  it  has  been 
possible  to  make  n-p-n  structures  with  base  layer  thicknesses  of  2  X  10~* 
cm.  The  uniformity  of  the  layers  in  a  given  specimen  is  better  than  ten 
per  cent  of  the  layer  thickness.  Fig.  1  illustrates  the  uniformity  of  the 
layers.  This  figure  is  an  enlarged  photograph  of  a  view  perpendicular 
to  the  surface  of  the  specimen.  A  bevel  which  makes  an  angle  of  five 
degrees  with  the  original  surface  has  been  polished  on  the  specimen.  This 
angle  magnifies  the  layer  thickness  by  11.5.  The  layer  is  defined  by  an 
etchant  which  preferentially  stains  p-type  silicon^  and  the  width  of  the 
layer  is  measured  with  a  calibrated  microscope. 

After  diffusion  the  entire  surface  of  the  silicon  wafer  is  covered  with 
the  diffused  n-  and  p-type  layers,  see  Fig.  2(a).  Electrical  contact  must 
now  be  made  to  the  three  regions  of  the  device.  The  base  contact  can 
be  made  by  polishing  a  bevel  on  the  specimen  to  expose  and  magnify 
the  base  layer  and  then  alloying  a  lead  to  this  region  by  the  same  tech- 


f.^  *f^'-  *; 


'>i 


i      *  /i 


n-TfPE  DIFFUSED  LAV^ER 
fo-t^^E*OiFFUSED    LAYER 


i»# 


OF^GIt^L  n-TYPE 
CRYSTAl. 


I 1  EQUIVALENT   TO  2  X  lO"'*  CM 

LAYER  THICKNESS 

Fig-  1  —  Angle  section  of  a  double  diffused  silicon  wafer.  The  p-type  center 
ayer  is  approximately  2  X  10-<  cm  thick. 


4  THE    BELL    SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 

niques  employed  in  the  fabrication  of  grown  junction  transistors.  Fig. 
2(b).  However,  a  much  simpler  technique  has  been  evolved.  If  the  sur- 
face concentration  of  the  donor  diffusant  is  maintained  below  a  certain 
critical  value,  it  is  possible  to  alloy  an  aluminum  wire  directly  through 
the  diffused  n-type  layer  and  thus  make  effective  contact  to  the  base 
layer,  Fig.  2(c).  Since  the  resistivity  of  the  original  silicon  wafer  is  one 
to  five  ohm-cm,  the  aluminum  will  be  rectifying  to  this  region.  It  has 
been  experimentally  shown  that  if  the  surface  concentration  of  the 
donor  diffusant  is  less  than  the  critical  value  mentioned  above,  the 
aluminum  will  also  be  rectifying  to  the  diffused  n-type  region  and  the 
contact  becomes  merely  an  extension  of  the  base  layer.  The  n-layers 
produced  by  diffusing  from  elemental  antimony  are  below  the  critical 
concentration  and  the  direct  aluminum  alloying  technique  is  feasible. 


-n  +  TYPE  DIFFUSED  LAYER 


-p-TYPE  DIFFUSED  LAYER 


n  + 


n+ 


-ALUMINUM  WIRE 

p  + ALUMINUM  DOPED 
REGROWTH  LAYER 


n-TYPE 


(b) 


,^- ALUMINUM  WIRE 

P  +  ALUMINUM   DOPED 
, REGROWTH  LAYER 


^M'nY  ^-i-r 


n-TYPE 


(c) 


Fig.  2  — ■  Schematic  illustralioii  of  (a)  double  diffused  n-p-n  wafer,  (b)  angle 
section  method  of  making  base  contact,  and  (c)  direct  alloying  method  of  making 
base  contact. 


DIFFUSED    EMITTER   AND    BASE    SILICON   TRANSISTORS 


AU-Sb  PLATED 
POINT 


VAPORIZED  Al 

LINE 
0.005  CM    WIDE 


t  MM 


Fig.  3  —  Mounted  double  diffused  transistor. 

Contact  to  the  emitter  layer  is  achieved  by  alloying  a  film  of  gold 
containing  a  small  amount  of  antimony.  Since  this  alloy  will  produce 
an  n-type  regrowth  layer,  it  is  only  necessary  to  insure  that  the  gold- 
antimony  film  does  not  alloy  through  the  p-type  base  layer,  thus  shorting 
the  emitter  to  the  collector.  This  is  controlled  by  limiting  the  amount  of 
gold-antimony  alloy  which  is  available  by  using  a  thin  evaporated  film 
or  by  electroplating  a  thin  film  of  gold-antimony  alloy  on  an  inert  metal 
point  and  alloying  this  structure  to  the  emitter  layer. 

Ohmic,  contact  to  the  collector  is  produced  by  alloying  the  silicon 
wafer  to  an  inert  metal  tab  plated  with  a  gold-antimony  alloy. 


6  THE   BELL   SYSTEM    TECHNICAL   JOURNAL,    JANUARY    1956 

The  transistors  whose  characteristics  are  reported  in  this  paper  were 
prepared  from  3  ohm-cm  n-type  siHcon  using  antimony  and  ahmiinum 
as  the  diffusants.  The  base  contact  was  produced  by  evaporating  alumi- 
num through  a  mask  so  that  a  hne  approximately  0.005  X  0.015  cm  in 

o 

lateral  dimensions  and  100,000  A  thick  was  formed  on  the  surface.  This 
aluminum  line  was  alloyed  through  the  emitter  layer  in  a  subsequent 
operation.  The  wafer  was  then  alloyed  onto  the  plated  kovar  tab.  A 
small  area  approximately  0.015  cm  in  diameter  was  masked  around  the 
line  and  the  wafer  was  etched  to  remove  the  unwanted  layers.  The  unit 
was  then  mounted  in  a  header.  Electrical  contact  to  the  collector  was 
made  by  soldering  to  the  kovar  tab.  Contact  to  the  base  was  made  with 
a  tungsten  point  pressure  contact  to  the  alloyed  aluminum.  Contact 
to  the  emitter  was  made  by  bringing  a  gold-antimony  plated  tungsten 
point  into  pressure  contact  with  the  emitter  layer.  The  gold-antimony 
plate  was  then  alloyed  by  passing  a  controlled  electrical  pulse  between 
the  plated  point  and  the  transistor  collector  lead.  Fig.  3  is  a  photograph 
of  a  mounted  unit. 

2.0   ELECTRICAL    CHARACTERISTICS 

The  frequency  cutoffs  of  experimental  double  diffused  silicon  tran- 
sistors fabricated  as  described  above  are  an  order  of  magnitude  higher 
than  the  known  cutoff  frequencies  of  earlier  silicon  transistors.  This  is 
shown  in  Fig.  4  which  gives  the  measured  common  base  and  common 
emitter  current  gains  for  one  of  these  units  as  a  function  of  frequency. 
The  common  base  short-circuit  current  gain  is  seen  to  have  a  cutoff  fre- 
quency of  about  120  mc/sec.  The  common  emitter  short-circuit  current 
gain  is  shown  on  the  same  figure.  The  low-freciuency  current  gain  is 
better  than  thirty  decibels  and  the  cutoff  frequency  which  is  indicated 
by  the  freciuency  at  which  the  gain  is  3  db  below  its  low-frequency 
value  is  3  mc/sec.  This  is  an  exceptionally  large  common  emitter  band- 
width for  a  thirty  db  common  emitter  current  gain  and  is  of  the  same 
order  of  magnitude  as  that  obtained  with  the  highest  frequency  ger- 
manium transistors  (e.q.,  p-n-i-p  or  tetrode)  which  had  been  made 
prior  to  the  diffused  base  germanium  transistor. 


^  Tlio  iiicroasp  in  (•oiiiinon  haso  current  gain  ahovc  unity  (indicated  by  current 
gain  in  decibels  being  positive)  in  the  vicinity  of  50  mc/sec  is  caused  by  a  reactance 
gain  error  in  the  common  base  measurement.  This  error  is  caused  by  a  combination 
of  the  emitter  to  ground  parasitic  capacitance  and  the  i)ositive  reactance  com- 
ponent of  the  transistor  input  impedance  resulting  from  phase  shift  in  the  ali)ha 
current  gain. 

'  C.  A.  Lee,  A  High-Frequency  Diffused  Base  Germanium  Transistor,  see 
page  23. 


DIFFUSED    EMITTER   AND    BASE    SILICON   TRANSISTORS 


z 
< 

o 

I- 

z 

LJ 

a. 
cr 

D 
O 


40 


30 


20 


(0 


-\0 


-20 


-30 


Ie  = 

3  MA 

Vc 

=  10  VOLTS 

COMMON^ 
EMITTER 

N 

'OCCB    —   ^  ^^ 
OCq   =   0.9716 

['=^"=106MC 

l-Ofg 

\     facb  =  i20MC 

\ 

COMMON 

BASE 

\ 

\ 

\ 

0.1     0.2       0.5     1.0      2  6        10      20         50     100    200 

FREQUENCY  IN  MEGACYCLES  PER  SECOND 


500  1000 


Fig.  4  — ■  Short-circuit  current  gain  of  a  double  diffused  silicon  n-p-n  transistor 
as  a  function  of  frequency  in  the  common  emitter  and  common  base  connections. 


Fig.  5  shows  a  high-freciueiicy  lumped  constant  equivalent  circuit 
for  the  double  diffused  silicon  transistor  whose  current  gain  cutoff  char- 
acteristic is  shown  in  Fig.  4.  External  parasitic  capacitances  have  been 
omitted  from  the  circuit.  The  configuration  is  the  conventional  one  for 
junction  transistors  with  two  exceptions.  A  series  resistance  rj  has  been 
added  in  the  emitter  circuit  to  account  for  contact  resistance  resulting 
from  the  fact  that  the  present  emitter  point  contacts  are  not  perfectly 
ohmic.  A  second  resistance  r/  has  been  added  in  the  collector  circuit  to 
account  for  the  ohmic  resistance  of  the  n-type  silicon  between  the  col- 
lector terminal  and  the  effective  collector  junction.  This  resistance  exists 
in  all  junction  transistors  but  in  larger  area  low  frequency  junction 
transistors  its  effect  on  alpha-cutoff  is  sufficiently  small  so  that  it  has 
been  ignored  in  equivalent  circuits  of  these  devices.  The  collector  RC 


Ce  =  TmmF 


Pq  -]AU) 


Cc  =  0.52//^F  r '  _  ,50  co 


Tg  =  150; 


a 


J^C( 


•Le 


'%=QOCO 


COMMON    BASE    CURRENT 
GAIN    CUT-OFF    FREQUENCY 


■  120  MC 


Ic  =  3  MA 
Vc  =  10  VOLTS 


Fig.  5  ~  High-frequency  lumped   constant  equivalent  circuit  for  a  double 
diffused  silicon  n-p-n  transistor. 


8 


THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


cutoff  caused  by  the  collector  capacitance  and  the  combined  collector 
body  resistance  and  base  resistance  is  an  order  of  magnitude  higher 
than  the  measured  alpha  cutoff  frequency  and  therefore  is  not  too  serious 
in  impairing  the  very  high-frecjuency  performance  of  the  transistor. 
This  is  due  to  the  low  capacitance  of  the  collector  junction  which  is 
seen  to  be  approximately  0.5  mmf  at  10  volts  collector  voltage.  The 
base  resistance  of  this  transistor  is  less  than  100  ohms  which  is  quite  low 
and  compares  very  favorably  with  the  best  low  frequency  transistors 
reported  previously. 

The  low-frequency  characteristics  of  the  double  diffused  silicon  tran- 
sistor are  very  similar  to  those  of  other  junction  transistors.  This  is  il- 
lustrated in  Fig.  6  where  the  static  collector  characteristics  of  one  of 
these  transistors  are  given.  At  zero  emitter  current  the  collector  current 
is  too  small  to  be  seen  on  the  scale  of  this  figure.  The  collector  current 


45 


40 


35 


30 


25 


20 


15 


10 


-5 


le=0 

2 

4            6 

8 

10 

12 

] 

J 

14/ 

^ 

J^ 

^ 

y^ 

^ 

2  4  6  8  10  12  14 

CURRENT,  If,  IN   MILUAMPERES 


Fig.  6  —  Collector  characteristics  of  a  double  diffused  silicon   n-p-n   tran- 
sistor. 


DIFFUSED    EMITTER   AND    BASE    SILICON   TRANSISTORS 


9 


0.98 


0.94 


0.90 


0.86 


a 


0.82 


0.78 


0.74 


0.70 


T=150°C, 

^ 

^ 

^ 

^ 

^ 

7 

<^ 

y 
^ 

^ 

^ 

\ 

/ 

9/ 

y 

24, 5M 
65-W 

/> 

7 

/24.5 

t35^y\ 

7 

15ol 

/ 

/ 

1 

1 

1 

_L. 

1 

1 

1 

,1 

0.1  0.2  0.4     0.6         1  2  4        6     8   10  20 

CURRENT,  Ig,  IN   MILLIAMPERES 

Fig.  7  —  Alpha  as  a  function  of  emitter  current  and  temperature  for  a  double 
diffused  silicon  n-p-n  transistor. 


under  this  condition  does  not  truly  saturate  but  collector  junction  re- 
sistance is  very  high.  Collector  junction  resistances  of  50  megohms  at 
reverse  biases  of  50  volts  are  common. 

The  continuous  power  dissipation  permissible  with  these  units  is  also 
shown  in  Fig.  6.  The  figure  shows  dissipation  of  200  milliwatts  and  the 
units  have  been  operated  at  400  milliwatts  without  damage.  As  illus- 
trated in  Fig.  3  no  special  provision  has  been  made  for  power  dissipation 
and  it  would  appear  from  the  performance  obtained  to  date  that  powers 
of  a  few  watts  could  be  handled  by  these  iniits  with  relatively  minor 
provisions  for  heat  dissipation.  However,  it  can  also  be  seen  from  Fig.  6 
that  at  low  collector  voltages  alpha  decreases  rapidly  as  the  emitter 
current  is  increased.  The  transistor  is,  therefore,  non-linear  in  this 
range  of  emitter  currents  and  collector  voltages.  In  many  applications, 
this  non-linearity  may  limit  the  operating  range  of  the  device  to  values 
below  those  which  would  be  permissible  from  the  point  of  view  of  con- 
tinuous power  dissipation. 

Fig.  7  gives  the  magnitude  of  alpha  as  a  function  of  emitter  current 
for  a  fixed  collector  voltage  of  10  volts  and  a  number  of  ambient  tem- 
peratures. These  curves  are  presented  to  illustrate  the  stability  of  the 
parameters  of  the  double  diffused  silicon  transistor  at  increased  ambient 
temperatures.  Over  the  range  from  1  to  15  milliamperes  emitter  current 
and  25°C  to  150°C  ambient  temperature,  alpha  is  seen  to  change  only 


10  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

by  approximately  2  per  cent.  This  amounts  to  only  150  parts  per  million 
change  in  alpha  per  degree  centigrade  change  in  ambient  temperature. 
The  decrease  in  alpha  at  low  emitter  currents  shown  in  Fig.  7  has  been 
observed  in  every  double  diffused  silicon  transistor  which  has  been  made 
to  date.  Although  this  effect  is  not  completely  understood  at  present  it 
could  be  caused  by  recombination  centers  in  the  base  layer  that  can 
be  saturated  at  high  injection  levels.  Such  saturation  would  result  in  an 
increase  in  effective  lifetime  and  a  corresponding  increase  in  alpha.  The 
large  increase  in  alpha  with  temperature  at  low  emitter  currents  is  con- 
sistent with  this  proposal.  It  has  also  been  observed  that  shining  a  strong 
light  on  the  transistor  will  produce  an  appreciable  increase  in  alpha  at 
low  emitter  currents  but  has  little  effect  at  high  emitter  currents.  A 
strong  light  would  also  be  expected  to  saturate  recombination  centers 
which  are  active  at  low  emitter  currents  and  this  behavior  is  also  con- 
sistent with  the  above  proposal. 

3.0   DISCUSSION    OF   THE   TRANSISTOR   STRUCTURE 

Although  the  low  frequency  electrical  characteristics  of  the  double 
diffused  silicon  transistor  which  are  presented  in  Section  2  are  quite 
similar  to  those  usually  obtained  in  junction  transistors,  the  structure 
of  the  double  diffused  transistor  is  sufficiently  different  from  that  of  the 
grown  junction  or  alloy  transistor  that  a  discussion  of  some  design 
principles  is  warranted.  This  section  is  devoted  to  a  general  discussion 
of  the  factors  which  determine  the  electrical  characteristics  of  the  tran- 
sistors. In  Section  4  the  general  ideas  of  Section  3  are  applied  in  a  more 
specialized  fashion  to  the  double  diffused  structure  and  a  detailed  cal- 
culation of  electrical  parameters  is  presented. 

One  essential  difference  between  the  double  diffused  transistor  and 
grown  junction  or  alloy  transistors  arises  from  the  manner  in  which  the 
impurities  are  distributed  in  the  three  active  regions.  In  the  ideal  case 
of  a  double-doped  grown  junction  transistor  or  an  alloy  transistor  the 
concentration  of  impurities  in  a  given  region  is  essentially  uniform  and 
the  transition  from  one  conductivity  type  to  another  at  the  emitter  and 
collector  junctions  is  abrupt  giving  rise  to  step  junctions.  On  the  other 
hand  in  the  double  diffused  structure  the  distribution  of  impurities  is 
more  closely  described  by  the  error  function  complement  and  the  emitter 
and  collector  junctions  are  graded.  Tlu\se  differences  can  have  an  appre- 
ciable influence  on  the  electrical  beha\'ior  of  the  transistors. 

Fig.  8(a)  shows  the  probable  distribution  of  donor  impurities,  No  , 
and  acceptor  impurities,  A''^  ,  in  a  double  diffused  n-p-n.  Fig.  8(b)  is  a 


DIFFUSED    EMITTER   AND    BASE    SILICON   TRANSISTORS 


11 


DONORS 

ACCEPTORS 


DISTANCE 

(a) 


DISTANCE    *• 

(b) 

Fig.  8  —  Diagrammatic  representation  of  (a)  donor  and  acceptor  distributions 
and  (b)  uncompensated  impuritj-  distribution  in  a  double  diffused  n-p-n  tran- 
sistor. 


plot  of  Nd  —  Na  which  would  result  from  the  distribution  in  Fig.  8(a). 
Kromer  has  shown  that  a  nonuniform  distribution  of  impurities  in  a 
semiconductor  will  produce  electric  fields  which  can  influence  the  flow 
of  electrons  and  holes.  For  example,  in  the  base  region  the  fields  between 
the  emitter  junction,  Xe ,  and  the  minimum  in  the  Nd  —  Na  curve,  x', 
will  retard  the  flow  of  electrons  toward  the  collector  while  the  fields 
between  this  minimum  and  the  collector  jvmction,  Xc ,  will  accelerate  the 
flow  of  electrons  toward  the  collector.  These  base  laj^er  fields  will  affect 
the  transit  time  of  minority  carriers  across  the  base  and  thus  contribute 

*  H.  Kromer,  On  Diffusion  and  Drift  Transistor  Theory  I,  II,  III,  Archiv.  der 
Electr.  Ubertragung,  8,  pp.  223-228,  pp.  363-369,  pp.  499-504,  1954. 


12  THE   BELL   SYSTEM   TECHNICAL   JOUENAL,    JANUARY    1956 

to  the  fre(iuency  response  of  the  transistor.  In  addition  the  base  re- 
sistance will  be  dependent  on  the  distribution  of  both  diffusants.  These 
three  factors  are  discussed  in  detail  below. 

Moll  and  Ross  have  determined  that  the  minority  current,  /,„  ,  that 
will  flow  into  the  base  region  of  a  transistor  if  the  base  is  doped  in  a  non- 
uniform manner  is  given  by 


f  N(x)  dx 


where  rii  is  the  carrier  concentration  in  intrinsic  material,  q  is  the  elec- 
tronic charge,  V  is  the  applied  voltage,  Dm  is  the  diffusion  coefficient  of 
the  minority  carriers,  and  the  integral  represents  the  total  number  of 
uncompensated  impurities  in  the  base.  The  primary  assumptions  in  this 
derivation  are  (1)  planar  junctions,  (2)  no  recombination  in  the  base 
region,  and  (3)  a  boundary  condition  at  the  collector  junction  that  the 
minority  carrier  density  at  this  point  equals  zero.  It  is  also  assumed  that 
the  minority  carrier  concentration  in  the  base  region  just  adjacent  to  the 
emitter  junction  is  equal  to  the  equilibrium  minority  carrier  density  at 
this  point  multiplied  by  the  Boltzman  factor  exp  (qV/kT).  It  is  of  special 
interest  to  note  that  Im  depends  only  on  the  total  number  of  uncom- 
pensated impurities  in  the  base  and  not  on  the  manner  in  which  they 
are  distributed. 

In  the  double  diffused  transistor,  it  has  been  convenient  from  the 
point  of  ease  of  fabrication  to  make  the  emitter  layer  approximately  the 
same  thickness  as  the  base  layer.  It  has  been  observed  that  heating  sili- 
con to  high  temperatures  degrades  the  lifetime  of  n-  and  p-type  silicon 
in  a  similar  manner.  Both  base  and  emitter  layers  have  experienced  the 
same  heat  treatment  and  to  a  first  approximation  it  can  be  assumed  that 
the  lifetime  in  the  two  regions  will  be  essentially  the  same.  Thus  as- 
sumptions (1)  and  (2)  should  also  apply  to  current  flow  from  base  to 
emitter.  If  we  assume  that  the  surface  recombination  \'elocity  at  the 
free  surface  of  the  emitter  is  infinite,  then  this  imposes  a  boundary 
condition  at  this  side  of  the  emitter  which  under  conditions  of  forward 
bias  on  the  emitter  is  equivalent  to  assumption  (3).  Thus  an  equation 
of  the  form  of  (3.1)  should  also  give  the  minority  current  flow  from  base 
to  emitter.  Since  the  emitter  efficiency,  y,  is  given  by 


^  J.  Tj.  Moll  and  I.  M.  Ross,  The  J)opendencc  of  Transistor  Paramotors  on  tlie 
Distribution  of  Base  Layer  liesistivity,  Proc.  I.R.E.  in  press. 
8  G.  Bemski,  private  comnmnication. 


DIFFUSED    EMITTER   AND    BASE    SILICON    TRANSISTORS  13 

/m  (emitter  to  base) 

-y     =    . . . 

/^(emitter  to  base)  +  /„j(base  to  emitter) 

proper  substitution  of  (3.1)  will  give  the  emitter  efficiency  of  the  double 
diffused  n-p-n  transistor, 

1 


7    = 


J-'n 


Z).^''^-^"^ 


dx 


p  .6  (3.2) 


\  (No  -  iVj  dx 


In  (3.2),  Dp  is  the  diffusion  coefficient  of  holes  in  the  emitter,  /)„  is  the 
diffusion  coefficient  of  electrons  in  the  base  and  the  ratio  of  integrals  is 
the  ratio  of  total  uncompensated  doping  in  the  base  to  that  in  the 
emitter. 

A  calculation  of  transit  time  is  more  difficult.  Kromer  has  studied 
the  case  of  an  aiding  field  which  reduces  transit  time  of  minority  carriers 
across  the  base  region  and  thus  increases  frequency  response.  In  the 
double  diffused  transistor  the  situation  is  more  complex.  Near  the 
emitter  side  of  the  base  region  the  field  is  retarding  (Region  R,  see  Fig.  8) 
and  becomes  aiding  (Region  A)  only  after  the  base  region  doping  reaches 
a  maximum.  The  case  of  retarding  fields  has  been  studied  by  Lee  and 
by  MoU.^  At  present,  the  case  for  a  base  region  containing  both  types  of 
fields  has  not  been  solved.  However,  at  the  present  state  of  knowledge 
some  speculations  about  transit  time  can  be  made. 

The  two  factors  of  primary  importance  are  the  magnitude  of  the 
built-in  fields  and  the  distance  over  which  they  extend.  In  the  double 
diffused  transistor,  the  widths  of  regions  R  and  A  are  determined  by  the 
surface  concentrations  and  diffusion  coefficients  of  the  diffusants.  It 
Can  be  shown  by  numerical  computation  that  if  region  R  constitutes  no 
more  than  30-40  per  cent  of  the  entire  base  layer  width,  then  the  overall 
effect  of  the  built-in  fields  will  be  to  aid  the  transport  of  minority  car- 
riers and  to  produce  a  reduction  in  transit  time.  In  addition  the  absolute 
magnitude  of  region  R  is  important.  If  the  point  x'  should  occur  within 
an  effective  Debye  length  from  the  emitter  junction,  i.e.,  if  x'  is  located 
in  the  space  charge  region  associated  with  the  emitter  junction,  then  the 
retarding  fields  can  be  neglected. 

The  base  resistance  can  also  be  calculated  from  surface  concentrations 
and  diffusion  coefficients  of  the  impurities.  This  is  done  by  considering 
the  base  layer  as  a  conducting  sheet  and  determining  the  sheet  con- 

'  J.  L.  Moll,  private  communication. 


14  THE    BELL    SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

ductivity  from  the  total  number  of  uncompensated  impurities  per  square 
centimeter  of  sheet  and  the  approjiriate  moliility  weighted  to  account 
for  impurity  scattering. 

4.0   CALCULATION    OF   DESIGN    PARAMETERS 

To  calculate  the  parameters  which  determine  emitter  efficiency,  transit 
time,  and  base  resistance  it  is  assumed  that  the  distribution  of  uncom- 
pensated impurities  is  given  by 

N(x)  =  Nicrfc  f  -  N-2erJc^  +  Nz  (4.1) 

where  A^i  and  A^2  are  the  surface  concentrations  of  the  emitter  and  base 
impurity  diffusants  respectively,  Li  and  L^  are  their  respective  diffusion 
lengths,  and  Nz  is  the  original  doping  of  the  semiconductor  into  which 
the  impurities  are  diffused.  The  impurity  diffusion  lengths  are  defined  as 

Li  =  2  V/M     and     L2  =  2  ^Ddo  (4.2) 

where  the  D's  are  the  respective  diffusion  coefficients  and  the  f's  are  the 
diffusion  times. 

Equation  (4.1)  can  be  reduced  to 


r(^)  =  Ti  erfc  I  -  Ta  erfc  X^  +  1  (4.3) 


where 


For  cases  of  interest  here,  r(^)  will  be  zero  at  two  points,  a  and  13, 
and  will  have  one  minimum  at  ^'.  In  the  transistor  structure  the  emitter 
junction  occurs  at  ^  =  ^v  and  the  collector  junction  occurs  at  ^  =  (3. 
Thus  the  base  width  is  determined  by  13  —  a.  The  extent  of  aiding  and 
retarding  fields  in  the  base  is  determined  by  ^'.  The  integral  of  (4.3) 
from  0  to  a,  I\  ,  and  from  o  to  ^,  I2 ,  are  the  integrals  of  interest  in  (3.2) 
and  thus  determine  emitter  efficiency.  In  addition  I2  is  the  integral  from 
which  base  resistance  can  be  calculated. 

The  calculations  which  follow  apply  only  for  values  of  ri/r2  and  To 
greater  than  ten.  Some  of  the  simplifying  assumptions  which  are  made 
will  not  apply  at  lower  values  of  these  parameters  where  the  distribution 
of  both  diffusants  as  well  as  the  background  doping  affect  the  structure 
in  all  three  regions  of  the  device. 


DIFFUSED    EMITTER   AND    BASE    SILICON   TRANSISTORS 


15 


4.1  Base  Width 

From  Fig.  8  and  (4.3)  it  can  be  seen  that  for  r2  ^  10,  a  is  essentially 
independent  of  r2  and  is  primarily  a  function  of  T1/T2  and  X.  Fig.  9  is  a 
plot  of  a  versus  ri/r2  with  X  as  the  parameter.  The  particular  plot  is  for 
r2  =  10  .  Although  as  stated  a  is  essentially  independent  of  r2 ,  at  lower 
values  of  r2,  a  may  not  exist  for  the  larger  values  of  X,  i.e.,  the  p-layer 
does  not  form. 

In  the  same  manner,  it  can  be  seen  that  ^  is  essentially  independent  of 
T]/T2  and  is  a  function  only  of  r2  and  X.  Fig.  10  is  a  plot  of  /3  versus  F^ 
with  X  as  a  parameter.  This  plot  is  for  Ti/Fo  =  10  and  at  larger  Fi/Fo , 
/3  may  not  exist  at  large  X. 


10" 


\0' 


10 


r2=)o'' 

/// 

// 

/ 

^ 

::i 

ll 

r     / 

/ 

m 

0/  / 

' 

> 

/os/ 

1 

i 

1 

/// 

'o.e/ 

/ 

f  0.7/ 

/ 

/// 

/ 

/ 

<.e 

I 

w. 

W 

/ 
/ 

/ 

1.0 


1.4 


1.8 


2.2  2.6 

a 


3.0 


3.4 


3.8 


Fig.  9  —  Emitter  layer  thickness  (in  reduced  units)  as  a  function  of  the  ratio 
of  the  surface  concentrations  of  the  diffusing  impurities  (ri/r2)  and  the  ratio  of 
their  diffusion  lengths  (X). 


16 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 


The  base  width 


W  =  ^  —  a 


can  be  obtained  from  Figs.  9  and  10.  a,  13  and  iv  can  be  converted  to 
centimeters  by  nuiltiplying  by  the  appropriate  value  of  Li  . 

4.2  Emitter  Efficiency 

With  the  hmits  a  and  /3  determined  above,  the  integrals  h  and  1 2  can 
be  calculated.  Examination  of  the  integrals  shows  that  h  is  closely  pro- 
portional to  ri/r2  and  also  to  r2 .  On  the  other  hand  I2  is  closely  propor- 
tional to  r2  and  essentially  independent  of  ri/r2 .  Thus,  the  ratio  of 
/2//1  which  determines  7  depends  primarily  on  ri/r2 .  Fig.  11  is  a  plot 
of  the  constant  /2//1  contours  in  the  ri/T2  —  X  plane  for  lo/h  ii^  the 
range  from  — 1.0  to  —0.01.  The  graph  is  for  r2  =  10  .  Since  from  (3.2) 


7   = 


1 


1  _  ^h 

Dnh 


(4.4) 


for  an  n-p-n  transistor,  and  assuming  Dp/Dn   =   /^  for  silicon,  then 


to' 


(0- 


10' 


10 


1' 

1 

\= 

..J\ 

0.6- 
0.5- 

::ffl 

M 

\u 

|6  In 
1     1° 

1 

\\\ 

( 

0.2 

0.1 

'/// 

/// 

0.01/ 

ill 

7 

/ 

/ 

/// 

/ 

/ 

/ 

10 


20  50 


100    200 


500     1000 


Fig.  10  —  (Collector  junction  dopth  (in  rodurod  units)  as  a  function  of  the  sur- 
face concuMit.ration  (in  reduced  units)  of  llie  dilfusaiit  wliicli  determines  the  con- 
ductivity type  of  the  l)ase  layer  (I'.')  and  liie  ratio  of  tlie  dilTusioii  lengths  (X)  of 
the  tAvo  diffusing  inii)urifies. 


DIFFUSED    EMITTER    AND    BASE    SILICON   TRANSISTORS 

10" 


17 


10 


H 

Ta 


10 


10 


r2  =  io'* 

2 

w 

\v 

V 

2 
? 

1 

^ 

1 

\\ 

\ 

t 

^. 

\ 

\I2/I 

1 

2 

V 

,\ 

N-0 

VO.05 

02 

-i.o\ 

-0.3S^ 

32X^0 

'\ 

0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


Fig.  11  — ^Dependence  of  emitter  efficiency  upon  diffusant  surface  concentra- 
tions and  diffusion  lengths.  The  lines  of  constant  /2//1  are  essentially  lines  of 
constant  emitter  efficiency.  The  ordinate  is  the  ratio  of  surface  concentrations  of 
the  two  diffusants  and  the  abscissa  is  the  ratio  of  their  diffusion  lengths. 

/2//1  =  — 1.0  corresponds  to  a  7  of  0.75  and  /2//1  =  —0.01  corresponds 
to  a  7  of  0.997. 


4.. 3  Base  Resistance 

It  was  indicated  above  that  I2  depends  principally  on  r2  and  X.  Fig.  12 
is  a  plot  of  the  constant  I2  contours  in  the  r2  —  X  plane  for  I2  in  the  range 
from  —10^  to  —10.  The  graph  is  for  Ti/To  =  10.  The  base  layer  sheet 
conductivity,  cjb  ,  can  be  calculated  from  these  data  as 


Qb  =   —qtihTjiNz 


(4.5) 


where  q,  L\  and  A^3  are  as  defined  above  and  /I  is  a  mobility  properly 
weighted  to  account  for  impurity  scattering  in  the  non-uniformly  doped 
base  region.  The  units  of  gb  are  mhos  per  square. 


18 


THE  BELL  SYSTEM  TECHNICAL  JOURNAL,  JANUARY   1956 


10- 

1 2= -10,00^ 

/ 

/ 

7/ 

1 

/ 

-5000/ 

r 

/  / 

// 

/ 

/ 

/       -1000/ 

// 

// 

/  / 1 

2 

1 

/ 

/-5oa 

/  / 

/   / 

/  1 

^/^^ 

/ 

/ 

/ 

/I 

^/ 

/  / 

1 1 

10 

/ 

/ 

// 

v. 

/-ioy 

V 

11 

/  / 

/ 

/, 

// 

/-/  , 

(I 

5 

// 

/, 

-^ 

/J 

/ 

V/ 

/ 

2 

^ 

^ 

^ 

f^ 

u 

10 

102 

r 

/  / 

^ 

/ 
/ 

^ 

5 

— 1 

0 

^/ 

V 

r 

10 

/ 

/ 

0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


Fig.  12  —  Dependence  of  base  layer  sheet  condiictivitj^  on  diffusant  surface 
concentrations  and  diffusion  lengths.  The  lines  of  constant  Ii  are  essentiallj'  lines 
of  constant  base  sheet  conductivity.  The  ordinate  is  the  surface  concentration 
(in  reduced  units)  of  the  diffusant  which  determines  the  conductivity  type  of  the 
base  layer  and  the  abscissa  is  the  ratio  of  the  diffusion  lengths  of  the  two  difi'using 
impurities. 

4.4  Transit  Time 

With  a  knowledge  of  where  the  minimum  value,  ^',  of  (4.3)  occurs, 
it  is  possible  to  calculate  over  what  fraction  of  the  base  width  the  fields 
are  retarding.  The  interesting  quantity  here  is 

13  -  a 

^  is  a  function  of  ri/r2  and  X  and  varies  only  very  slowly  with  ri/r2 . 
a  is  also  a  function  of  ri/r2  and  X  and  varies  only  slowly  with  ri/r2 . 
The  most  rapidly  changing  part  of  bJi  is  l^  which  depends  primarily  on 
r2  as  noted  above.  Fig.  13  is  a  plot  of  the  constant  LR  contours  in  the 
r2  —  X  plane  for  values  of  A/2  in  the  range  0.1  to  0.3.  This  graph  is 


DIFFUSED    EMITTER   AND    BASE    SILICON   TRANSISTORS 


19 


lor  data  with  ri/r2  =  10.  As  ri/r2  increases  at  constant  r2  and  X,  AR 
decreases  slightly.  At  ri/r2  =  10\  the  average  change  in  AR  is  a  decrease 
of  about  25  per  cent  for  constant  r2  and  X  when  AR  ^  0.3.  The  error  is 
larger  for  values  of  AR  greater  than  0.3.  It  was  noted  above  that  when 
AR  becomes  greater  than  0.3,  the  retarding  fields  become  dominant. 
Therefore,  this  region  is  of  slight  interest  in  the  design  of  a  high  frequency 
transistor. 

4.5  A  Sample  Design 

By  superimposing  Figs.  11,  12  and  13  the  ranges  of  r2 ,  ri/r2  and  X 
which  are  consistent  with  desired  values  of  y,  gt  and  AR  can  be  deter- 


0.7 


Fig.  1.3  —  Dependence  of  the  built-in  field  distribution  on  concentrations  and 
diffusion  lengths.  The  lines  of  constant  aR  indicate  the  fraction  of  the  base  layer 
thickness  over  which  built-in  fields  are  retarding.  The  ordinate  is  the  surface 
concentration  (in  reduced  units)  of  the  diffusant  which  determines  the  conductiv- 
ity type  of  the  base  layer  and  the  abscissa  is  the  ratio  of  the  diffusion  lengths  of 
the  two  diffusing  impurities. 


20  THE  BELL  SYSTEM  TECHNICAL  JOURNAL,  JANUARY   1956 

mined  by  the  area  enclosed  by  the  specified  contour  lines.  It  is  also 
possible  to  compare  the  measured  parameters  of  a  specific  device  and 
observe  how  closely  they  agree  with  what  is  predicted  from  the  estimated 
concentrations  and  diffusion  coefficients.  This  is  done  below  for  the 
transistor  described  in  Sections  1  and  2. 

The  comparison  is  complicated  by  the  fact  that  the  exact  values  of  the 
surface  concentrations  and  diffusion  coefficients  are  not  known  {Precisely 
enough  at  present  to  permit  an  accurate  evaluation  of  the  design  theory. 
However,  the  following  values  of  concentrations  and  diffusion  coefficients 
are  thought  to  be  realistic  for  this  transistor. 

iVi  =  5  X  10^'        /)i  =  3  X  10"''  /i  =  5.7  X  lO' 

iV2  =  4  X  10''        Di  =  2.5  X  10""         t^=  1.2  X  lO' 

Nz  =  10'' 

From  these  values  it  is  seen  that 

Ti/ra  =  12.5;         r,  =  400;         X  =  0.6 

From  Fig.  9,  a  =  1.9  and  from  Fig.  10,  /3  =  3.6  and  therefore  w  =  1.7. 
Measurement  of  the  emitter  and  base  layer  dimensions  showed  that  these 
layers  were  approximately  the  same  thickness  which  was  3.8  X  10"  cm. 
Thus  the  ifieasured  ratio  of  emitter  width  to  base  width  of  unity  is  in 
good  agreement  with  the  ^'alue  of  1.1  predicted  from  the  assumed  con- 
centrations and  diffusion  coefficients. 

From  Fig.  11,  lo/h  ~  —0.01.  If  this  value  is  substituted  into  (4.4), 
7  =  0.997.  This  compares  with  a  measured  maximum  alpha  of  0.972. 

From  Fig.  12,  lo  =  —15.  Assuming  an  average  hole  mobility  of  350 
cm' /volt.  sec.  and  evaluating  Li  from  the  measured  emitter  thickness 
and  the  calculated  a,  (4.5)  gives  a  value  of  gb  =  1.7  X  10^  mhos  per 
square.  The  geometry  of  the  emitter  and  base  contacts  as  shown  in  Fig. 
3  makes  it  difficult  to  calculate  the  effective  base  resistance  from  the 
sheet  conductivity  even  at  very  small  emitter  currents.  In  addition  at 
the  very  high  inje{;tion  levels  at  which  these  transistors  are  operated  the 
calculation  of  effective  base  resistance  becomes  very  difficult.  However, 
from  the  geometr}^  it  would  be  expected  that  the  effective  base  re- 
sistance would  l)c  no  greater  than  0.1  of  the  sheet  resistivity  or  600  ohms. 
This  is  about  seven  times  larger  than  the  measured  \'alue  of  80  ohms 
reported  in  Section  2. 

From  Fig.  b3,  A/^  is  approximately  0.20.  Thus  there  should  be  an  over- 
all aiding  elfect  of  the  built-in  fields.  In  addition  the  impurity  gradient 
at  the  emitter  junction  is  believed  to  be  approximately  lO'Vcm  and  the 


DIFFUSED    EMITTER   AND    BASE    SILICON    TRANSISTORS  21 

space  charge  associated  with  this  gradient  will  extend  approximately 
2   X   10  ■'  cm  into  the  base  region.  The  base  thickness  over  which  re- 
tarding fields  extend  is  AR  times  the  base  width  or  7.6  X  10~^  cm.  Thus 
the  first  quarter  of  region  R  will  be  space  charge  and  can  be  neglected. 
The  frequency  cutoff  from  pure  diffusion  transit  is  given  by 

2A3D  ,.    , 

where  W  is  the  measured  base  layer  thickness.  Assuming  D  —  25  cmVsec 
for  electrons  in  the  base  region,  ,/'„  =  (w  mc/sec.  Since  the  measured 
cutoff  was  120  mc/sec,  the  predicted  aiding  effect  of  the  built-in  field 
is  evidently  present. 

These  computations  illustrate  how  the  measured  electrical  parameters 
can  be  used  to  check  the  values  of  the  surface  concentrations  and  dif- 
fusion coefficients.  Conversely  knowledge  of  the  concentrations  and 
diffusion  coefficients  aid  in  the  design  of  devices  which  will  have  pre- 
scribed electrical  parameters.  The  agreement  in  the  case  of  the  transistor 
described  above  is  not  perfect  and  indicates  errors  in  the  proposed  values 
of  the  concentrations  and  diffusion  coefficients.  However,  it  is  sufficiently 
close  to  be  encouraging  and  indicate  the  value  of  the  calculations. 

The  discussion  of  design  has  been  limited  to  a  very  few  of  the  important 
parameters.  Junction  capacitances,  emitter  and  collector  resistances  are 
among  the  other  important  characteristics  which  have  been  omitted 
here.  Presumably  all  of  these  quantities  can  be  calculated  if  the  detailed 
structure  of  the  device  is  known  and  the  structure  should  be  susceptible 
to  the  type  of  analysis  used  above.  Another  fact,  which  has  been  ignored, 
is  that  these  transistors  were  operated  at  high  injection  levels  and  a  low 
level  analysis  of  electrical  parameters  was  used.  All  of  these  other  factors 
must  be  considered  for  a  detailed  understanding  of  the  device.  The  object 
of  this  last  section  has  been  to  indicate  one  path  which  the  more  detailed 
analysis  might  take. 

5.0   CONCLUSIONS 

By  means  of  multiple  diffusion,  it  has  been  possible  to  produce  silicon 
transistors  with  alpha-cutoff  above  100  mc/sec.  Refinements  of  the 
described  technicjues  offer  the  possibility  of  even  higher  frequency  per- 
formance. These  transistors  show  the  other  advantages  expected  from 
silicon  such  as  low  saturation  currents  and  satisfactory  operation  at 
high  temperatures. 

The  structure  of  the  double  diffused  transistor  is  susceptible  to  design 


22  THE  BELL  SYSTEM  TECHNICAL  JOURNAL,  JANUARY   1956 

analysis  in  a  fashion  similar  to  that  which  has  been  applied  to  other  junc- 
tion transistors.  The  non-uniform  distribution  of  impurities  produces 
significant  electrical  effects  which  can  be  controlled  to  enhance  appre- 
cial)ly  the  high-frequency  behavior  of  the  devices. 

The  extreme  control  inherent  in  the  use  of  diffusion  to  distribute  im- 
purities in  a  semiconductor  structure  suggests  that  this  technique  will 
become  one  of  the  most  valuable  in  the  fabrication  of  semiconductor 
devices. 

ACKNOWLEDGEMENT 

The  authors  are  indebted  to  several  people  who  contributed  to  the 
work  described  in  this  paper.  In  particular,  the  double  diffused  silicon 
from  which  the  transistors  were  prepared  was  supplied  by  C.  S.  Fuller 
and  J.  A.  Ditzenberger.  The  data  on  diffusion  coefficients  and  concen- 
trations were  also  obtained  by  them. 

P.  W.  Foy  and  G.  Kaminsky  assisted  in  the  fabrication  and  mounting 
of  the  transistors  and  J.  M.  Klein  aided  in  the  electrical  characterization. 
The  computations  of  the  various  solutions  of  the  diffusion  equation,  (4.3), 
were  performed  by  Francis  Maier.  In  addition  many  valuable  discussions 
with  C.  A.  Lee,  G.  Weinreich,  J.  L.  Moll,  and  G.  C.  Dacey  helped  formu- 
late many  of  the  ideas  presented  herein. 


A  High-Frequency  Diffused  Base 
Gernianiuni  Transistor 

By  CHARLES  A.  LEE 

(Manuscript  received  November  15,  1955) 

Techniques  of  impurity  diffusion  and  alloying  have  been  developed  which 
make  possible  the  construction  of  p-n-p  junction  transistors  utilizing  a 
diffused  surface  layer  as  a  base  region.  An  important  Jeature  is  the  high 
degree  of  dimensional  control  obtainable.  Diffusion  has  the  advantages  of 
being  able  to  produce  uniform  large  area  junctions  which  may  be  utilized  in 
high  power  devices,  and  very  thin  surface  layers  which  may  be  utilized  in 
high-frequency  devices. 

Transistors  have  been  made  in  germanium  which  typically  have  alphas 
of  0.98  and  alpha-cutoff  frequencies  of  500  mcls.  The  fabrication,  electrical 
characterization,  and  design  considerations  of  these  transistors  are  dis- 
cussed. 

INTRODUCTION 

Recent  work  ■  concerning  diffusion  of  impurities  into  germanium 
and  silicon  prompted  the  suggestion  that  the  dimensional  control  in- 
herent in  these  processes  be  utilized  to  make  high-frecjuency  transistors. 

One  of  the  critical  dimensions  of  junction  transistors,  which  in  many 
cases  seriously  restricts  their  upper  freciuency  limit  of  operation,  is  the 
thickness  of  the  base  region.  A  considerable  advance  in  transistor  proper- 
ties can  be  accomplished  if  it  is  possible  to  reduce  this  dimension  one  or 
two  orders  of  magnitude.  The  diffusion  constants  of  ordinary  donors 
and  acceptors  in  germanium  are  such  that,  with'n  realizable  tempera- 
tures and  times,  the  depth  of  diffused  surface  layers  may  be  as  small  as 
10"  cm.  Already  in  the  present  works  layers  slightly  less  than  1  micron 
(10~  cm)  thick  have  been  made  and  utilized  in  transistors.  Moreover, 
the  times  and  temperatures  required  to  produce  1  micron  surface  laj^ers 
permit  good  control  of  the  depth  of  penetration  and  the  concentration 
of  the  diffusant  in  the  surface  layer  with  techniciues  described  below. 

If  one  considers  making  a  transistor  whose  base  region  consists  of  such 

23 


24  THE  BELL  SYSTEM  TECHNICAL  JOURNAL,  JANUARY   1956 

a  diffused  surface  layer,  several  problems  become  immediately  apparent : 

(1)  Control  of  body  resistivity  and  lifetime  during  the  diffusion  heat- 
ing cycle. 

(2)  Control  of  the  surface  concentration  of  the  diffusant. 

(3)  INIaking  an  emitter  on  the  surface  of  a  thin  diffused  layer  and 
controlling  the  depth  of  penetration. 

(4)  Making  an  ohmic  base  contact  to  the  diffused  surface  layer. 
One  approach  to  the  solution  of  these  problems  in  germanium  which  has 
enabled  us  to  make  transistors  with  alpha-cutoff  frequencies  in  excess 
of  500  mc/sec  is  described  in  the  main  body  of  the  paper. 

An  important  characteristic  feature  of  the  diffusion  technique  is  that 
it  produces  an  impurity  gradient  in  the  base  region  of  the  transistor. 
This  impurity  gradiant  produces  a  "built-in"  electric  field  in  such  a 
direction  as  to  aid  the  transport  of  minority  carriers  from  emitter  to 
collector.  Such  a  drift  field  may  considerably  enhance  the  frequency 
response  of  a  transistor  for  given  physical  dimensions. 

The  capabilities  of  these  new  techniques  are  only  partially  realized 
by  their  application  to  the  making  of  high  frequency  transistors,  and 
even  in  this  field  their  potential  has  not  been  completely  explored.  For 
example,  with  these  techniques  applied  to  making  a  p-n-i-p  structure 
the  possibility  of  constructing  transistor  amplifiers  with  usable  gain  at 
frequencies  in  excess  of  1,000  mc/sec  now  seems  feasible. 

DESCRIPTION   OF  TRANSISTOR  FABRICATION  AND   PHYSICAL  CHARACTERIS- 
TICS 

As  starting  material  for  a  p-n-p  structure,  p-type  germanium  of  0.8 
ohm-cm  resistivity  was  used.  From  the  single  crystal  ingot  rectangular 
bars  were  cut  and  then  lapped  and  polished  to  the  approximate  dimen- 
sions: 200  X  60  X  15  mils.  After  a  slight  etch,  the  bars  were  washed  in 
deionized  water  and  placed  in  a  vacuum  oven  for  the  diffusion  of  an 
n-type  impurity  into  the  surface.  The  vacuum  oven  consisted  of  a  small 
molybdenum  capsule  heated  by  radiation  from  a  tungsten  coil  and  sur- 
rounded by  suitable  radiation  shields  made  also  of  molybdenum.  The 
capsule  could  be  baked  out  at  about  1,900°C  in  order  that  impurities 
detrimental  to  the  electrical  characteristics  of  the  germaniinn  be  evapo- 
rated to  sufficiently  low  levels. 

As  a  source  of  n-type  impurity  to  be  placed  with  the  p-type  bars  in 
the  molybdenum  oven,  arsenic  doped  germanium  was  used.  The  rela- 
tively high  vapor  pressure  of  the  arsenic  was  reduced  to  a  desirable  range 
(about  lO"*  nun  of  Ilg)  by  diluting  it  in  germanium.  The  use  of  ger- 
manium eliminated  any  additional  problems  of  contamination  by  the 


A  HIGH-FREQUENCY  DIFFUSED   BASE  GERMANIUM  TRANSISTOR         25 

dilutant,  and  provided  a  convenient  means  of  determining  the  degree  of 
dilution  by  a  measurement  of  the  conductivity.  The  arsenic  concentra- 
tions used  in  the  source  crystal  were  typically  of  the  order  of  10  '-10^^/cc. 
These  concentrations  were  rather  high  compared  to  the  concentrations 
desired  in  the  diffused  surface  layers  since  compensation  had  to  be  made 
for  losses  of  arsenic  due  to  the  imperfect  fit  of  the  cover  on  the  capsule 
and  due  to  some  chemical  reaction  and  adsorption  which  occurred  on  the 
internal  surfaces  of  the  capsule. 

The  layers  obtained  after  diffusion  were  then  evaluated  for  sheet  con- 
ductivity and  thickness.  To  measure  the  sheet  conductivity  a  four-point 
probe  method^  was  used.  An  island  of  the  surface  layer  was  formed  by 
masking  and  etching  to  reveal  the  junction  between  the  surface  layer 
and  the  p-type  body.  The  island  was  then  biased  in  the  reverse  direction 
with  respect  to  the  body  thus  effectively  isolating  it  electrically  during 
the  measurement  of  its  sheet  conductivity.  The  thickness  of  the  surface 
layer  was  obtained  by  first  lapping  at  a  small  angle  to  the  original  surface 
(3^-2°~l°)  and  locating  the  junction  on  the  beveled  surface  with  a  thermal 
probe;  then  multiplying  the  tangent  of  the  angle  between  the  two  sur- 
faces by  the  distance  from  the  edge  of  the  bevel  to  the  junction  gives  the 
desired  thickness.  Another  particularly  convenient  method  of  measuring 
the  thickness'  is  to  place  a  half  silvered  mirror  parallel  to  the  original  sur- 
face and  count  fringes,  of  the  sodium  D-Yme  for  example,  from  the  edge 
of  the  bevel  to  the  junction.  Typically  the  transistors  described  here 
were  prepared  from  diffused  layers  with  a  sheet  conductivity  of  about 
200  ohms/square,  and  a  layer  thickness  of  (1.5  ±  0.3)  X  10~   cm. 

When  the  surface  layer  had  been  evaluated,  the  emitter  and  base  con- 
tacts were  made  using  techniques  of  vacuum  evaporation  and  alloying. 

o 

For  the  emitter,  a  film  of  aluminum  approximately  1,000  A  thick  was 
evaporated  onto  the  surface  through  a  mask  which  defined  an  emitter 
area  of  1  X  2  mils.  The  bar  with  the  evaporated  aluminum  was  then 
placed  on  a  strip  heater  in  a  hydrogen  atmosphere  and  momentarily 
brought  up  to  a  temperature  sufficient  to  alloy  the  alimiinum.  The 
emitter  having  been  thus  formed,  the  bar  was  again  placed  in  the  masking 
jig  and  a  film  of  gold-antimony  alloy  from  3,000  to  4,000  A  thick  was 
evaporated  onto  the  surface.  This  film  was  identical  in  area  to  the 
emitter,  and  was  placed  parallel  to  and  0.5  to  1  mil  away  from  the 
emitter.  The  bar  was  again  placed  on  the  heater  strip  and  heated  to  the 
gold-germanium  eutectic  temperature,  thus  forming  the  ohmic  base 
contact.  The  masking  jig  was  constructed  to  permit  the  simultaneous 
evaporation  of  eight  pairs  of  contacts  on  each  bar.  Thus,  using  a  3-mil 
diamond  saw,  a  bar  could  be  cut  into  eight  units. 


20 


THE  BELL  SYSTEM  TECHNICAL  JOURNAL,  JANUARY    1956 


Each  unit,  with  an  alloyed  emitter  and  base  contact,  was  then  soldered 
to  a  platinum  tab  with  indium,  a  sufficient  quantity  of  indium  being- 
used  to  alloy  through  the  n-type  surface  layer  on  the  back  of  the  unit. 
One  of  the  last  steps  was  to  mask  the  emitter  and  base  contacts  with  a 
6-  to  8-mil  diameter  dot  of  wax  and  form  a  small  area  collector  junction 
by  etching  the  unit  attached  to  the  platinum  tab,  in  CP4.  After  washing 
in  solvents  to  remove  the  wax,  the  unit  was  mounted  in  a  header  designed 
to  allow  electrolytically  pointed  wire  contacts  to  be  made  to  the  base  and 
emitter  areas  of  the  transistor.  These  spring  contacts  were  made  of  1-mil 
phosphor  bronze  wire. 

ELECTRICAL    CHARACTERIZATION 

Of  the  parameters  that  characterize  the  performance  of  a  transistor, 
one  of  the  most  important  is  the  short  circuit  current  gain  (alpha)  ver- 
sus frequency.  The  measured  variation  of  a  and  q:/(1  —  a)  (short-circuit 
current  gain  in  the  grounded  emitter  circuit)  as  a  function  of  frequency 
for  a  typical  unit  is  shown  in  Fig.  1 .  For  comparison  the  same  parameters 
for  an  exceptionally  good  unit  are  shown  in  Fig.  2. 

In  order  that  the  alpha-cutoff  frequency  be  a  measure  of  the  transit 
time  of  minority  carriers  through  the  active  regions  of  the  transistor,  any 
resistance-capacity  cutoffs,  of  the  emitter  and  collector  circuits,  must  lie 
considerably  higher  than  the  measured  /„  .  In  the  emitter  circuit,  an 
external  contact  resistance  to  the  aluminum  emitter  of  the  order  of  10 


U1 

_J 

LU 

eg 

o 

lij 

Q 


•4U 

( 
30 

20 

>-( 

— , 

4.3 

MC 

UNIT  0-3    p- 

n-p 

Ge 

Ie  =  2  MA 
Vc  =-10  VOLTS 
ao=  0.982 

'     1 

s 

S.  1-a 

6  DB 
OCT/> 

PER  ^' 

VE 

■> 

^s 

1  0 

0 

-10 

l«l 

w 

> 

\ 

46 

3  M( 

1 

; 

^ 

* 

0.1  0.2         0.4   0,6        1  2  4       6     8  10  20  40     60       100        200        400  1000 

FREQUENCY    IN    MEGACYCLES    PER    SECOND 


Fig.  1  —  The  grounded  emitter  and  grounded  base  response  versus  frequency 
for  a  typical  unit. 


A  HIGH-FREQUENCY  DIFFUSED   BASE  GERMANIUM  TRANSISTOR         27 


40 


30 


10 

_l 

LU 

5   20 
o 

LJJ 

Q 


10 


o- 

^ 

« 

3.4  M 

C 

UNIT  M-2  p-r 

Ie  =  2MA 

1-p 

Ge 

N 

-—    oc 
1      \-oc 

Vc=-10  VOLTS 
OCo-  0.980 

6Db\ 
PER  A 
OCTAVE 

^N 

^'s 

oc 

i-C 

v^       540  MC 

^\ 

\ 

-10 

0.1         0.2         0.4  0.6       1  2  4       6    e  10  20  40    60     100        200       400  1000 

FREQUENCY    IN    MEGACYCLES    PER     SECOND 

Fig.  2  —  The  grounded  emitter  and  grounded  base  response  versus  frequency 
for  an  exceptionally  good  unit. 

to  20  ohms  and  a  junction  transition  capacity  of  1  fx^xid  were  measured. 
The  displacement  current  which  flows  through  this  transition  capacity 
reduces  the  emitter  efficiency  and  must  be  kept  small  relative  to  the 
injected  hole  current.  With  1  milliampere  of  ciu"rent  flowing  through  the 
emitter  junction,  and  conseciuently  an  emitter  resistance  of  26  ohms, 
I  the  emitter  cutoff  for  this  transistor  was  above  6,000  mc/sec.  One  can 
now  see  that  the  emitter  area  must  be  small  and  the  current  density 
high  to  attain  a  high  emitter  cutoff  freciuency.  The  fact  that  a  low  base 
resistance  requires  a  high  level  of  doping  in  the  base  region,  and  thus  a 
high  emitter  transition  capacity,  restricts  one  to  small  areas  and  high 
current  densities. 

In  the  collector  circuit  capacities  of  0.5  to  0.8  ^l^xid  at  a  collector  volt- 
age of  — 10  volts  were  measured.  There  was  a  spreading  resistance  in  the 
collector  body  of  about  100  ohms  which  was  the  result  of  the  small 
emitter  area.  The  base  resistance  was  approximately  100  ohms.  If  the 
phase  shift  and  attenuation  due  to  the  transport  of  minority  carriers 
through  the  base  region  w^ere  small  at  the  collector  cutoff  frequency,  the 
(effective  base  resistance  would  be  decreased  by  the  factor  (1  —a).  The 
collector  cutoff  frequency  is  then  given  by 


where  Cc  =  collector  transition  capacity 

and     Re  =  collector  body  spreading  resistance. 


28  THE  BELL  SYSTEM  TECHNICAL  JOURNAL,  JANUARY   1956 

However,  in  the  transistors  described  here  the  base  region  produces  the 
major  contribution  to  the  observed  alpha-cutoff  frequency  and  it  is  more 
appropriate  to  use  the  expression 


2irCcin  +  Re) 


where  n  =  base  resistance.  This  cutoff  frequency  could  be  raised  by  in- 
creasing the  collector  voltage,  but  the  allowable  power  dissipation  in  the 
mounting  determines  an  upper  limit  for  this  voltage.  It  should  b  noted 
that  an  increase  in  the  doping  of  the  collector  material  would  raise  the 
cutoff  since  the  spreading  resistance  is  inversely  proportional  to  Na  , 
while  the  junction  capacity  for  constant  collector  voltage  is  only  pro- 
portional  to  Na    . 

The  low-frequency  alpha  of  the  transistor  ranged  from  0.95  to  0.99 
with  some  exceptional  units  as  high  as  0.998.  The  factors  to  be  con- 
sidered here  are  the  emitter  efficiency  y  and  the  transport  factor  (3. 
The  transport  factor  is  dependent  upon  the  lifetime  in  the  base  region, 
the  recombination  velocity  at  the  surface  immediately  surrounding  the 
emitter,  and  the  geometry.  The  geometrical  factor  of  the  ratio  of  the 
emitter  dimensions  to  the  base  layer  thickness  is  >  10,  indicating  that 
solutions  for  a  planar  geometry  may  be  assumed.^  If  a  lifetime  in  the  base 
region  of  1  microsecond  and  a  surface  recombination  velocity  of  2,000 
cm/sec  is  assumed  a  perturbation  calculation  gives 

iS  =  0.995 

The  high  value  of  ^  obtained  with  what  is  estimated  to  be  a  low  base 
region  lifetime  and  a  high  surface  recombination  velocity  indicates  that 
the  observed  low  frecjuency  alpha  is  most  probably  limited  by  the 
emitter  injection  efficiency.  As  for  the  emitter  injection  efficiency,  within 
the  accuracy  to  which  the  impurity  concentrations  in  the  emitter  re- 
growth  layer  and  the  base  region  are  known,  together  with  the  thick- 
nesses of  these  two  regions,  the  calculated  efficiency  is  consistent  with 
the  experimentally  observed  values. 

Considerations  of  Transit  Time 

An  examination  of  what  agreement  (^xists  between  the  alpha-cutoff 
frequency  and  the  physical  measurements  of  the  base  region  involves 
the  me(;hanism  of  transport  of  minority  carriers  through  the  active 
regions  of  the  transistor.  The  "active  regions"  include  the  space  charge 


A  HIGH-FREQUENCY  DIFFUSED   BASE  GERMANIUM  TRANSISTOR         29 

region  of  the  collector  junction.  The  transit  time  through  this  region 
is  no  longer  a  negligible  factor.  A  short  calculation  will  show  that  with 
—  10  volts  on  the  collector  junction,  the  space  charger  layer  is  about 
4  X  10"^  cm  thick  and  that  the  frequency  cutoff  associated  with  trans- 
port through  this  region  is  approximately  3,000  mc/sec. 

The  remaining  problem  is  the  transport  of  minority  carriers  through 
the  base  region.  Depending  upon  the  boundary  conditions  existing  at  the 
surface  of  the  germanium  during  the  diffusion  process,  considerable 
gradients  of  the  impurity  density  in  the  surface  layer  are  possible.  How- 
ever, the  problem  of  what  boundary  conditions  existed  during  the  diffu- 
sion process  employed  in  the  fabrication  of  these  transistors  w^ill  not  be 
discussed  here  because  of  the  many  uncertainties  involved.  Some  quali- 
tative idea  is  necessary  though  of  how  electric  fields  arising  from  impurity 
gradients  may  affect  the  frequency  behavior  of  a  transistor  in  the  limit 
of  low  injection. 

If  one  assumes  a  constant  electric  field  as  would  result  from  an  ex- 
ponential impurity  gradient  in  the  base  region  of  a  transistor,  then  the 
continuity  eciuation  may  be  solved  for  the  distribution  of  minority 
carriers.*  From  the  hole  distribution  one  can  obtain  an  expression  for 
the  transport  factor  j3  and  it  has  the  form 


/3  =  e" 


r?  sinh  Z  -{-  Z  cosh  Z 


where 


1,    Ne       IqE 
^"2^^iV;  =  2^^' 

z  ^  [i^  +  ,r' 

IV' 

Ne  =  donor  density  in  base  region  at  emitter  junction 
Nc  =  donor  density  in  base  region  at  collector  junction 

E  =  electric  field  strength 
Dp  =  diffusion  constant  for  holes 

w  =  width  of  the  base  layer 


30 


THE  BELL  SYSTEM  TECHNICAL  JOURNAL,  JANUARY   1956 


A  plot  of  this  function  for  various  values  of  rj  is  shown  in  Fig.  3.  For  ??  =  0, 
the  above  expression  reduces  to  the  well  known  case  of  a  uniformly  doped 
base  region.  The  important  feature  to  be  noted  in  Fig.  3  is  that  relatively 
small  gradients  of  the  impurity  distribution  in  the  base  layer  can  produce 
a  considerable  enhancement  of  the  frequency  response. 

It  is  instructive  to  calculate  what  the  alpha-cutoff  f  recjuency  would  be 
for  a  base  region  with  a  uniform  distribution  of  impurity.  The  effective 
thickness  of  the  base  layer  may  be  estimated  by  decreasing  the  measured 
thickness  of  the  surface  layer  by  the  penetration  of  the  space  charge 
region  of  the  collector  and  the  depth  of  the  alloyed  emitter  structure. 
Using  a  value  for  the  diffusion  constant  of  holes  in  the  base  region  appro- 
priate to  a  donor  density  of  about  10  Vcc, 

300  mc/s  ^fa^  800  mc/s 

This  result  implies  that  the  frecjuency  enhancement  due  to  "built-in" 
fields  is  at  most  a  factor  of  two.  In  addition  it  was  observed  that  the 
alpha-cutoff  frequency  was  a  function  of  the  emitter  current  as  shown 
in  Fig.  4.  This  variation  indicates  that  at  least  intermediate  injection 


<Si 


£L 

'^     ^     77  siNhZ  +Z  coshz 
Z=(L5z5+772)'/2 

0.8 
0.6 

0.4 

'       > 

*~ 

:;^;~->i^ 

k.^ 

^ 

Nv 

^ 

N 

"\ 

V 

\ 

\ 

\ 

V 

^ 

\ 

\ 

>v, 

0.2 

A 

\ 

\ 

K 

\ 

\ 

\ 

i 

\ 

K 

\      \ 

\ 

0.08 

- 

^ 

\— 

^ 

A 

— \ — 

v\- 

0.06 
0.04 

- 

^^ 

\ 

^, 

^ 

\ 

^ 

1\ 

4i 

r 

0.02 

\ 

\ 

\ 

V 

\ 

\ 

V 

0.01 

1 

1 

1 

\ 

\ 

1 

1 

> 

1 

1 

1 

1 

_L 

0.1 


0.2 


0.4      0.6  0.8    1 


6      8     10 


20 


40       60    80  100 


w2 
<^-U}  -g-  ,  (RADIANS) 


Fig.  .3  —  The  variation  of  |  i3  |  ver.sii.s  frequency  for  various  values  of  a  uniform 
drift  field  in  the  base  region. 


A  HIGH-FREQUENCY  DIFFUSED  BASE  GERMANIUM  TRANSISTOR         31 


in 

_i 

LU 

m 
o 

LU 

a 

z 


b 

n 

=7^" 

'^^-^^ 

S— 1' 

i 

f 

\ 

^                      ' 

' 

; Q       ■   ■_;;;; -t 

Fv 

Rl 

k 

-5 

UNIT  0-3  p-n-p  Ge 

o      Ie  =  2  MA 
A      Ie=0.8MA 
D      Ig=0.4MA 

\ 

k^ 

^ 

\ 

\ 

\ 

10 

Vc 

=  -K 

)  VOLTS 

1 

1 

\ 

1 

1 

1 

10 


20 


30       40      50    60       80     100  200  300     400 

FREQUENCY    IN    MEGACYCLES    PER    SECOND 


600     800   1000 


Fig.  4 
current. 


The  variation  of  the  alpha-cutoff  frequency  as  a  function  of  emitter 


levels  exist  in  the  range  of  emitter  current  shown  in  Fig.  4.  The  conclu- 
sion to  be  drawn  then  is  that  electric  fields  produced  by  impurity 
gradients  in  the  base  region  are  not  the  dominant  factor  in  the  transport 
of  minority  carriers  in  these  transistors. 

The  emitter  current  for  a  low  level  of  injection  could  not  be  deter- 
mined by  measuring  /„  versus  /«  because  the  high  input  impedance  at 
very  low  levels  was  shorted  by  the  input  capacity  of  the  header  and 
socket.  Thus  at  very  small  emitter  currents  the  measured  cutoff  fre- 
quency was  due  to  an  emitter  cutoff  and  was  roughly  proportional  to 
the  emitter  current.  At  /e  ^  1  ma  this  effect  is  small,  but  here  at  least 
intermediate  levels  of  injection  already  exist. 

A  further  attempt  to  measure  the  effect  of  any  "built-in"  fields  by 
turning  the  transistor  around  and  measuring  the  inverse  alpha  proved 
fruitless  for  two  reasons.  The  unfavorable  geometrical  factor  of  a  large 
collector  area  an  a  small  emitter  area  as  well  as  a  poor  injection  effi- 
ciency gave  an  alpha  of  only 


a 


=  0.1 


Secondly,  the  injection  efficiency  turns  out  in  this  case  to  be  proportional 
to  oT^^'^  giving  a  cutoff  freciuency  of  less  than  1  mc/sec.  The  sciuare-root 
dependence  of  the  injection  efficiency  on  freciuency  may  be  readily  seen. 
The  electron  current  injected  into  the  collector  body  may  be  expressed  as 


Je  =  qDnN 


1    -)-    iu^Te 


1/2 


where  q  =  electronic  charge 


32  THE  BELL  SYSTEM  TECHNICAL  JOURNAL,  JANUARY   1956 

Dn  ^  diffusion  constant  of  electrons 

Vi  =  voltage  across  collector  junction 

Tic  =  density  of  electrons  on  the  p-type  side  of  the  collector  junction 

Te  =  lifetime  of  electrons  in  collector  body 

Le  =  diffusion  length  of  electrons  in  the  collector  body 

Since  the  inverse  cutoff  frequency  is  well  below  that  associated  with  the 
base  region,  we  may  regard  the  injected  hole  current  as  independent  of 
the  frequency  in  this  region.  The  injection  efficiency  is  low  so  that 

7  ;^  ^  «  1 

J  e 


Thus  at  a  frequency  where 


then 


cor, 


»1 


I 


-1/2 

An  interesting  feature  of  these  transistors  was  the  very  high  current 
densities  at  which  the  emitter  could  be  operated  without  appreciable  loss 
of  injection  efficiency.  Fig.  5  shows  the  transmission  of  a  50  millimicro- 
second pulse  up  to  currents  of  18  milliamperes  which  corresponds  to  a 
current  density  of  1800  amperes/cm".  The  injection  efficiency  should 
remain  high  as  long  as  the  electron  density  at  the  emitter  edge  of  the 
base  region  remains  small  compared  to  the  acceptor  density  in  the 
emitter  regrowth  layer.  When  high  injection  levels  are  reached  the  in- 
jected hole  density  at  the  emitter  greatly  exceeds  the  donor  density  in  th(> 
base  region.  In  order  to  preserve  charge  neutrality  then 

p  ^  n 

where  p  =  hole  density 

n  =  electron  density 

As  the  inject(Hl  hole  density  is  raised  still  further  the  electron  density 
will  eventually  become  comparable  to  the  acceptor  density  in  the 
emitter  regrowth  layer.  Tlie  density  of  acceptors  in  the  emitter  regrowth 


A  HIGH-FREQUENCY  DIFFUSED  BASE  GERMANIUM  TRANSISTOR         33 


30  46  60  75  90 

TIME     IN     MILLIMICROSECONDS 


>" 


0 
9 

"^ 

V 

4 

'^ 

\^ 

/ 

18 

V 

/ 

-15 


15 


30  45  60  75  90 

TIME     IN     MILLIMICROSECONDS 


105 


120 


136 


Fig.  5  —  Transmission  of  a  50  millimicrosecond  pulse  at  emitter  currents  up 
to  18  ma  by  a  typical  unit.  (Courtesy  of  F.  K.  Bowers). 

region  is  of  the  order  of 

and  this  is  to  be  compared  with  injected  hole  density  at  the  base  region 
iside  of  the  emitter  junction.  The  relation  between  the  injected  hole 
density  and  the  current  density  may  be  approximated  by^ 


J. 


w 


where  pi  =  hole  density  at  emitter  side  of  base  region 

w  =  width  of  base  region 

jA  short  calculation  indicates  that  the  emitter  efficiency  should  remain 
'high  at  a  current  density  of  an  order  of  magnitude  higher  than  1,800 
|amp/cm'.  The  measurements  were  not  carried  to  higher  current  densities 
jbecause  the  voltage  drop  across  the  spreading  resistance  in  the  collector 
was  producing  saturation  of  the  collector  junction. 

CONCLUSIONS 

Impurity  diffusion  is  an  extremely  powerful  tool  for  the  fabrication 
of  high  frequency  transistors.  Moreover,  of  the  50-odd  transistors  which 


34  THE  BELL  SYSTEM  TECHNICAL  JOURNAL,  JANUARY   1956 

were  made  in  the  laboratory,  the  characteristics  were  remarkably  uni- 
form considering  the  ^•ariations  usually  encountered  at  such  a  stage  of 
development.  It  appears  that  diffusion  process  is  sufficiently  controllable 
that  the  thickness  of  the  base  region  can  be  reduced  to  half  that  of  the 
units  described  here.  Therefore,  with  no  change  in  the  other  design 
parameters,  outside  of  perhaps  a  different  mounting,  units  with  a  1000 
mc/s  cutoff  frequency  should  be  possible. 

ACKNOWLEDGMENT 

The  author  wishes  to  acknowledge  the  help  of  P.  W.  Foy  and  W.  Wieg- 
mann  who  aided  in  the  construction  of  the  transistors,  D.  E.  Thomas  who 
designed  the  electrical  equipment  needed  to  characterize  these  units, 
and  J.  Klein  who  helped  with  the  electrical  measurements.  The  numerical 
evaluation  of  alpha  for  drift  fields  was  done  by  Lillian  Lee  whose  as- 
sistance is  gratefully  acknowledged. 

REFERENCES 

1.  C.  S.  Fuller,  Phys.  Rev.,  86,  pp.  136-137,  1952. 

2.  J.  Saby  and  W.  C.  Dunlap,  Jr.,  Phys.  Rev.,  90,  p.  630,  1953. 

3.  W.  Shocklej',  private  communication. 

4.  H.  Kromer,  Archiv.  der  Elek.  tlbertragung,  8,  No.  5,  pp.  223-228,  1954. 

5.  R.  A.  Logan  and  M.  Schwartz,  Phys.  Rev.,  96,  p.  46,  1954 

6.  L.  B.  Valdes,  Proc.  I.R.E.,  42,  pp.  420-427,  1954. 

7.  W.  L.  Bond  and  F.  M.  Smits,  to  be  published. 

8.  E.  S.  Rittner,  Pnys.  Rev.,  94,  p.  1161,  1954. 

9.  W.  M.  Webster,  Proc.  I.R.E.,  42,  p.  914,  1954. 

10.  J.  M.  Early,  B.S.T.J.,  33,  pp.  517-533,  1954. 


Waveguide  Investigations  with 
Millimicrosecond  Pulses 

By  A.  C.  BECK 

(Manuscript  received  October  11,  1955) 

Pulse  techniques  have  been  used  for  many  waveguide  testing  'puryoses. 
The  importance  of  increased  resolution  hy  means  of  short  pulses  has  led  to 
the  development  of  equipment  to  generate,  receive  and  display  pidses  about 
5  or  6  millimicroseconds  lo7ig.  The  equipment  is  briefly  described  and  its 
resolution  and  measuring  range  are  discussed.  Domi7ia7it  mode  waveguide 
and  antenna  tests  are  described,  and  illustrated.  Applications  to  midtimode 
waveguides  are  then  considered.  Mode  separation,  delay  distortion  and  its 
equalization,  and  mode  conversion  are  discussed,  and  examples  are  given. 
The  resolution  obtained  with  this  equipment  provides  information  that  is 
difficult  to  get  by  any  other  means,  and  its  use  has  proved  to  be  very  helpfid 
in  ivaveguide  investigations. 

CONTENTS 

1 .  Introduction 35 

2.  Pulse  Generation 36 

3.  Receiver  and  Indicator 41 

4.  Resolution  and  Measuring  Range 42 

5.  Dominant  Mode  Waveguide  Tests 43 

6.  Testing  Antenna  Installations 45 

7.  Separation  of  Modes  on  a  Time  Basis 48 

8.  Delay  Distortion 52 

9.  Delay  Distortion  Ecjualization 54 

10.  Measuring  Mode  Conversion  from  Isolated  Sources 57 

11.  Measuring  Distril)uted  Mode  Conversion  in  1  ong  Waveguides 61 

12.  Concluding  Remarks 65 

1.    INTRODUCTION 

Pulse  testing  techniques  have  been  employed  to  advantage  in  wave- 
guide investigations  in  numerous  ways.  The  importance  of  better  resolu- 
tion through  the  use  of  short  pulses  has  always  been  apparent  and,  from 
the  first,  eciuipment  was  employed  which  used  as  short  a  pulse  as  pos- 
sible. Radar-type  apparatus  using  magnetrons  and  a  pulse  width  of 
about  one-tenth  microsecond  has  seen  considerable  use  in  waveguide 
research,  and  many  of  the  results  have  been  published.'  •  - 

35 


36  THE   BELL    SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 

To  improve  the  resolution,  work  was  initiated  some  time  ago  by  S.  E. 
Miller  to  obtain  measuring  equipment  which  would  operate  with  much 
shorter  pulses.  As  a  result,  pulses  about  5  or  6  millimicroseconds  long 
became  available  at  a  frequency  of  9,000  mc.  In  a  pulse  of  this  length 
there  are  less  than  100  cycles  of  radio  frequency  energy,  and  the  signal 
occupies  less  than  ten  feet  of  path  length  in  the  transmission  medium. 
The  RF  bandwidth  required  is  about  500  mc.  In  order  to  obtain  such 
bandwidths,  traveling  wave  tubes  were  developed  by  J.  R.  Pierce  and 
members  of  the  Electronics  Research  Department  of  the  Laboratories. 
The  completed  amplifiers  were  designed  by  W.  W.  Mumford.  N.  J. 
Pierce,  R.  W.  Dawson  and  J.  W.  Bell  assisted  in  the  design  and  construc- 
tion phases,  and  G.  D.  Mandeville  has  been  closely  associated  in  all  of 
this  work. 

2.    PULSE    GENERATION 


These  millimicrosecond  pulses  have  been  produced  by  two  different 
types  of  generators.  In  the  first  equipment,  a  regenerative  pulse  gener- 
ator of  the  type  suggested  by  C.  C.  Cutler  of  the  Laboratories  was  used.^ 
This  was  a  very  useful  device,  although  somewhat  complicated  and  hard 
to  keep  in  adjustment.  A  brief  description  of  it  will  permit  comparisons 
with  a  simpler  generator  which  was  developed  a  little  later. 

A  block  diagram  of  the  regenerative  pulse  generator  is  shown  in  Fig.  1. 
The  fundamental  part  of  the  system  is  the  feedback  loop  drawn  with 
heavy  lines  in  the  lower  central  part  of  the  figure.  This  includes  a  travel- 
ing wave  amplifier,  a  waveguide  delay  line  about  sixty  feet  long,  a  crystal 
expander,  a  band-pass  filter,  and  an  attenuator.  This  combination  forms 
an  oscillator  which  produces  very  short  pulses  of  microwave  energy. 
Between  pulses,  the  expander  makes  the  feedback  loop  loss  too  high  for 
oscillation.  Each  time  the  pulse  circulates  around  the  loop  it  tends  to 
shorten,  due  to  the  greater  amplification  of  its  narrower  upper  part 
caused  by  the  expander  action,  until  it  uses  the  entire  available  band 
width.  A  500-mc  gaussian  band-pass  filter  is  used  in  the  feedback  loop,^ 
of  this  generator  to  determine  the  final  bandwidth.  An  automatic  gain 
control  operates  with  the  expander  to  limit  the  pulse  amplitude,  thus 
preventing  amplifier  compression  from  reducing  the  available  expansion. 

To  get  enough  separation  between  outgoing  pulses  for  reflected  pulse 
measurements  with  waveguides,  the  repetition  rate  would  need  to  be 
too  low  for  a  practical  delay  fine  length  in  the  loop.  Therefore  a  r2.8-mc 
fundamental  rate  was  chosen,  and  a  gated  traveling  wave  {\\\)v  ampfifier 
was  used  to  reduce  it  to  a  100-kc  rate  at  the  output.  This  amplifier  is 
kept  in  a  cutoff  condition  for  127  pulses,  and  then  a  gate  pulse  restores 


I 


i 


t 


WAVEGUIDE   TESTING   WITH    MILLIMIf'ROSECOND    PULSES 


37 


it  to  the  normal  amplifying  condition  for  fifty  millimicroseconds,  during 
which  time  the  128th  pulse  is  passed  on  to  the  output  of  the  generator 
as  shown  on  Fig.  1. 

The  synchronizing  system  is  also  shown  on  Fig.  1.  A  100-kc  quartz 
crystal  controlled  oscillator  with  three  cathode  follower  outputs  is  the 
basis  of  the  system.  One  output  goes  through  a  seven  stage  multiplier 
to  get  a  12.8-mc  signal,  which  is  used  to  control  a  pulser  for  synchroniz- 
ing the  circulating  loop.  Another  output  controls  the  gate  pulser  for  the 
output  traveling  wave  amplifier.  Accurate  timing  of  the  gate  pulse  is 
obtained  by  adding  the  12.8-mc  pulses  through  a  buffer  amplifier  to  the 
gate  pulser.  The  third  output  synchronizes  the  indicator  oscilloscope 
sweep  to  give  a  steady  pattern  on  the  screen. 

Although  this  equipment  was  fairly  satisfactory  and  served  for  many 


OSCILLATOR 

AND    CATHODE 

FOLLOWERS 

100  KC 


I  1 


MULTIPLIER 

100 KC  TO 

12.8  MC 


SYNC 

PULSER 

0.02  A  SEC 

12.8  MC 


500  MC 

BANDPASS 

FILTER 


GATE 

PULSER 

0.05  USEC 

100  KC 


A 


BUFFER 
AMPLIFIER 


"1 


CRYSTAL 
EXPANDER 


U 


AGC   I 


WAVEGUIDE 

DELAY 

LINE 


TW  TUBE 


■Y^ 


MILLI/iSEC/ 
9000  MC/' 
PULSES 
12.8  MC    RATE 


MlLLIyUSEC 

9000  MC 

PULSES 

100  KC    RATE 


GATED 
TW   TUBE 


SYNC   SIGNAL  TO 
INDICATOR  SCOPE 


Fig.  1  —  Block  diagram  of  the  regenerative  pulse  generator. 


38  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 

testing  purposes,  it  was  rather  complex  and  there  were  some  problems 
in  its  construction  and  use.  It  was  difficult  to  obtain  suitable  microwave 
crystals  to  match  the  waveguide  at  low  levels  in  the  expander.  Tliis 
would  make  it  even  more  difficult  to  build  this  type  of  pulse  generator 
for  higher  frequency  ranges.  Stability  also  proved  to  be  a  problem.  The 
frequency  multiplier  had  to  be  very  well  constructed  to  avoid  phase 
shift  due  to  drifting.  The  gate  pulser  also  required  care  in  design  and 
construction  in  order  to  get  a  stable  and  flat  output  pulse.  It  was  some- 
what troublesome  to  keep  the  gain  adjusted  for  proper  operation,  and 
the  gate  pulse  time  adjustment  required  some  attention.  The  pulse 
frequency  could  not  be  changed.  For  these  reasons,  and  in  order  to  get 
a  smaller,  lighter  and  less  complicated  pulse  generator,  work  was  carried 
out  to  produce  pulses  of  about  the  same  length  by  a  simpler  method. 

If  the  gated  output  amplifier  of  Fig.  1  were  to  have  a  CW  instead  of  a 
pulsed  input,  a  pulse  of  microwave  energy  would  nevertheless  appear  at 
the  output  because  of  the  presence  of  the  gating  pulse.  This  gating  pulse 
is  applied  to  the  beam  forming  electrode  of  the  tube  to  obtain  the  gating 
action.  If  the  beam  forming  electrode  could  be  pulsed  from  cutoff  to  its 
normal  operating  potential  for  a  very  short  time,  very  short  pulses  of 
output  energy  could  be  obtained  from  a  continuous  input  signal.  How- 
ever, it  is  difficult  to  obtain  millimicrosecond  video  gating  pulses  of  suf- 
ficient amplitude  for  this  purpose  at  a  100-kc  repetition  rate. 

A  traveling-wave  tube  amplifies  normally  only  when  the  helix  is 
within  a  small  voltage  range  around  its  rated  dc  operating  value.  For 
voltages  either  above  or  below  this  range,  the  tube  is  cut  off.  When  the 
helix  voltage  is  raised  through  this  range  into  the  cutoff  region  beyond 
it,  and  then  brought  back  again,  two  pulses  are  obtained,  one  during  a 
small  part  of  the  rise  time  and  the  other  during  a  small  part  of  the  return 
time.  If  the  rise  and  fall  times  are  steep,  very  short  pulses  can  be 
obtained.  Fig.  2  shows  the  pulse  envelopes  photographed  from  the 
indicator  scope  screen  when  this  is  done.  For  the  top  trace,  the  helix  was 
biased  300  volts  negatively  from  its  normal  operating  potential,  then 
pulsed  to  its  correct  operating  range  for  about  80  millimicroseconds, 
during  which  time  normal  amplification  of  the  CW  input  signal  was  ob- 
tained. The  effect  of  further  increasing  the  helix  video  pulse  amplitude 
in  the  positive  direction  is  shown  by  the  succeeding  lower  traces.  The 
envelope  dips  in  the  middle,  then  two  separated  pulses  remain  —  one 
during  a  part  of  the  rise  time  and  one  during  a  part  of  the  fall  time  of 
helix  voltage.  The  pulses  shown  on  the  bottom  trace  have  shortened  to 
about  six  millimicroseconds  in  length.  The  helix  pulse  had  a  positive 
amplitude  of  about  500  volts  for  this  trace. 


1 


WAVEGUIDE    TESTIXG    WITH    MILUMICROSErOXD    PULSES 


39 


Since  only  one  of  these  pulses  can  be  used  to  get  the  desired  repetition 
rate,  it  is  necessary  to  eliminate  the  other  pulse.  This  is  done  in  a  simi- 
lar manner  to  that  used  for  gating  out  the  undesired  pulses  in  the  re- 
generative pulse  generator.  However,  it  is  not  necessary  to  use  another 
amplifier,  as  was  required  there,  since  the  same  tube  can  be  used  for 
this  purpose,  as  well  as  for  producing  the  microwave  pulses.  Its  beam 
forming  electrode  is  biased  negatively  about  250  volts  with  respect  to 
the  cathode,  and  then  is  pulsed  to  the  normal  operating  potential  for 
about  50  millimicroseconds  during  the  time  of  the  first  short  pulse  ob- 
tained by  gating  the  helix.  Thus,  the  beam  forming  electrode  potential 
has  been  returned  to  the  cutoff  value  during  the  second  helix  pulse, 
which  is  therefore  eliminated. 
Il  A  block  diagram  of  the  resulting  double-gated  pulse  generator  is 
shown  in  Fig.  3.  Comparison  with  Fig.  1  shows  that  it  is  simpler 
than  the  regenerative  pulse  generator,  and  it  has  also  proved  more 
satisfactory  in  operation.  It  can  be  used  at  any  frequency  where  a  sig- 
nal source  and  a  traveling-wave  amplifier  are  available,  and  the  pulse 


Fig.  2  —  Envelopes  of  microwave  pulses  at  the  output  of  a  traveling  wave  am- 
lifier  with  continuous  wave  input  and  helix  gating.  The  gating  voltage  is  higher 
or  the  lower  traces. 


40 


THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


frequency  can  be  set  anywhere  within  the  bandwidth  of  the  travehng- 
wave  ampUfier  by  tuning  the  klystron  oscillator. 

The  pulse  center  frequency  is  shifted  from  that  of  the  klystron  os- 
cillator frequency  by  this  helix  gating  process.  An  over-simphfied  but 
helpful  explanation  of  this  effect  can  be  obtained  by  considering  that 
the  microwave  signal  voltage  on  the  helix  causes  a  bunching  of  the  elec- 
tron stream.  This^  bunching  has  the  same  periodicity  as  the  microwave 
signal  voltage  when  the  dc  helix  potential  is  held  constant.  However, 
since  the  helix  voltage  is  continuously  increased  in  the  positive  direction 
during  the  time  of  the  first  pulse,  the  average  velocity  of  the  last  bunches 
of  electrons  becomes  higher  than  that  of  the  earlier  bunches  in  the  pulse, 
because  the  later  electrons  come  along  at  the  time  of  higher  positive 
helix  voltage.  This  tends  to  shorten  the  total  length  of  the  series  of 
bunches,  resulting  in  a  shorter  w^avelength  at  the  output  end  of  the 
helix  and  therefore  a  higher  output  microwave  frequency.  On  the  second 
pulse,  obtained  when  the  helix  voltage  returns  toward  zero,  the  process 
is  reversed,  the  bunching  is  stretched  out,  and  the  frequency  is  de- 
creased. This  second  pulse  is,  however,  gated  out  in  this  arrangement 
by  the  beam-forming  electrode  pulsing  voltage.  The  result  for  this 
particular  tube  and  pulse  length  is  an  effective  output  frequency  ap- 
proximately 150  mc  higher  than  the  oscillator  frequency,  but  this  figure 
is  not  constant  over  the  range  of  pulse  frequencies  available  within  the 
amplifier  bandwidth. 


OSCILLATOR   AND 

CATHODE    FOLLOWERS 

100  KC 


KLYSTRON 

OSCILLATOR 

9000  MC 


BEAM    FORMING 

ELECTRODE 

PULSER 


HELIX 
PULSER 


^ 


PULSED 
TW  TUBE 


MILLI/aSEC 
9000  MC 
PULSES 


SYNC    SIGNAL  TO 
INDICATOR   SCOPE 


Fig.  3  —  Block  diugram  of  the  double-gated  traveling  wave  tube  millimicro- 
second pulse  generator. 


WAVEGUIDE   TESTING   WITH   MILLIMICROSECOND    PULSES 


41 


3.   RECEIVER  AND  INDICATOR 


The  receiving  equipment  is  shown  in  Fig.  4.  It  uses  two  traveUng- 
wave  amplifiers  in  cascade.  A  wide  band  detector  and  a  video  amplifier 
then  follow,  and  the  signal  envelope  is  displayed  by  connecting  it  to 
the  vertical  deflecting  plates  of  a  5  XP  type  oscilloscope  tube.  The 
video  amplifier  now  consists  of  two  Hewlett  Packard  wide  band  dis- 
tributed amplifiers,  having  a  baseband  width  of  about  175  mc.  The 
second  one  of  these  has  been  modified  to  give  a  higher  output  voltage. 
The  sweep  circuits  for  this  oscilloscope  have  been  built  especially  for 
this  use,  and  produce  a  sweep  speed  in  the  order  of  6  feet  per  micro- 
second. An  intensity  pulser  is  used  to  eliminate  the  return  trace.  These 
parts  of  the  system  are  controlled  by  a  synchronizing  output  from  the 
pulse  generator  100-kc  oscillator.  A  precision  phase  shifter  is  used  at 
the  receiver  for  the  same  purpose  that  a  range  unit  is  employed  in  radar 
systems.  This  has  a  dial,  calibrated  in  millimicroseconds,  which  moves 
the  position  of  a  pulse  appearing  on  the  scope  and  makes  accurate 
measurement  of  pulse  delay  time  possible. 

Fig.  4  also  shows  the  appearance  of  the  pulses  obtained  with  this 
equipment.  The  pulse  on  the  left-hand  side  of  this  trace  came  from  the 


PULSE 

SIGNAL 

9000  MC 


SYNC 

SIGNAL 
100  KC 


TW   TUBES 


VIDEO 
AMPLIFIER 


INTENSITY 

PULSER 

0.05/USEC 

100  KC 


PRECISION 

PHASE 

SHIFTER 


SWEEP 
GENERATOR 


DOUBLE-GATED 
PULSE 


REGENERATIVE 
PULSE 


Fig.  4  —  Block  diagram  of  millimicrosecond  pulse  receiver  and  indicator.  The 
idicator  trace  photograph  shows  pulses  from  each  type  of  generator. 


42 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 


newer  double-gated  pulse  generator,  while  the  pulse  on  the  right  was 
produced  by  the  regenerative  pulse  generator.  It  can  be  seen  that  they 
appear  to  have  about  the  same  pulse  width  and  shape.  This  is  partly 
due  to  the  fact  that  the  video  amplifier  bandwidth  is  not  c^uite  adequate 
to  show  the  actual  shape,  since  in  both  cases  the  pulses  are  slightly 
shorter  than  can  be  correctly  reproduced  through  this  amplifier.  The 
ripples  on  the  base  line  following  the  pulses  are  also  due  to  the  video 
amplifier  characteristics  when  used  with  such  short  pulses. 

4.    RESOLUTION  AND  MEASURING  RANGE 

Fig.  5  shows  a  piece  of  equipment  which  was  placed  between  the  pulse 
generator  and  the  receiver  to  show  the  resolution  which  can  be  obtained. 
This  waveguide  hybrid  junction  has  its  branch  marked  1  connected  to 
the  pulse  generator  and  branch  3  connected  to  the  receiver.  If  the  two 
side  branches  marked  2  and  4  were  terminated,  substantially  no  energy 
would  be  transmitted  from  the  pulser  straight  through  to  the  receiver. 
However,  a  short  circuit  placed  on  either  side  branch  will  send  energy 
through  the  system  to  the  receiver.  Two  short  circuits  were  so  placed 
that  the  one  on  branch  4  was  4  feet  farther  away  from  the  hybiid  junc- 
tion than  the  one  on  branch  2.  The  pulse  appearing  first  is  produced  l)y 
a  signal  traveling  from  the  pulse  generator  to  the  short  circuit  on  branch 
2  and  then  through  to  the  receiver,  as  shown  by  the  path  drawn  with 
short  dashes.  A  second  pulse  is  produced  by  the  signal  which  travels 


BRANCH 
2 


SHORT 
CIRCUIT 


BRANCH 


FROM 
PULSER 


TO 
RECEIVER 


FIRST   PULSE    PATH 
SECOND    PULSE    PATH 


SHORT 

CIRCUIT 


DOUBLE-GATED    PULSES 


REGENERATIVE    PULSES 


Fig.  5  —  W;iv(!guicle  hyhriil  ciicuil-  uscxl  to  demonstrate  resululion  of  milli- 
microsecond pulses.  Trace  photographs  of  pulses  from  each  type  of  generator  ;iie 
shown. 


WAVEGUIDE   TESTING   WITH   MILLIMICROSECOND    PULSES 


43 


TO  RECEIVER 
\ 


TE°    IN   3"DIAM  copper  GUIDE    (ISO  FT   LONG) 


Fig.  6  —  Waveguide  arrangement  and  oscilloscope  trace  photos  showing  pres- 
ence and  location  of  defective  joint.  The  dominant  mode  (TEn)  was  used  with  its 
polarization  changed  90  degrees  for  the  two  trace  photos. 

from  the  pulse  generator  through  branch  4  to  the  short  circuit  and  then 
to  the  receiver  as  shown  by  the  long  dashed  line.  This  pulse  has  traveled 
8  feet  farther  in  the  waveguide  than  the  first  pulse.  This  would  be  equiva- 
lent to  seeing  separate  radar  echoes  from  two  targets  about  4  feet  apart. 
Resolution  tests  made  in  this  way  \vith  the  pulses  from  the  regenerative 
pulse  generator,  and  from  the  double-gated  pulse  generator,  are  shown 
on  Fig.  5.  With  our  video  amplifier  and  viewing  equipment,  there  is 
no  appreciable  difference  in  the  resolution  obtained  using  either  type 
of  pulse  generator. 

The  measuring  range  is  determined  by  the  power  output  of  the  gated 
amplifier  at  saturation  and  by  the  noise  figure  of  the  first  tube  in  the 
receiver.  In  this  equipment  the  saturation  level  is  about  1  watt,  and  the 
noise  figure  of  the  first  receiver  tube  is  rather  poor.  As  a  result,  received 
pulses  about  70  db  below  the  outgoing  pulse  can  be  observed,  which  is 
I  enough  range  for  many  measurement  purposes. 


5.   DOMINANT  MODE  WAVEGUIDE  TESTS 

Fig.  6  shows  the  use  of  this  equipment  to  test  3'^  round  waveguides 
such  as  those  installed  between  radio  repeater  equipment  and  an  an- 
tenna. This  particular  150-foot  line  had  very  good  soldered  joints  and  was 
thought  to  be  electrically  very  smooth.  The  signal  is  sent  in  through  a 
transducer  to  produce  the  dominant  TEn  mode.  The  receiver  is  con- 
nected through  a  directional  coupler  on  the  sending  end  to  look  for  any 


44  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 


Fig.  7  —  Defective  joint  caused  by  imperfect  soldering  which  gave  the  reflec- 
tion shown  on  Fig.  6. 

reflections  from  imperfections  in  the  line.  The  overloaded  signal  at  the 
left  of  the  oscilloscope  trace  is  produced  by  leakage  directly  through 
the  directional  coupler.  The  overloaded  signal  on  the  other  end  of  this 
trace  is  produced  by  the  reflection  from  the  short  circuit  piston  at  the 
far  end  of  the  waveguide.  The  signal  between  these  two,  which  is  about 
45  db  down  from  the  input  signal,  is  produced  by  an  imperfect  joint 
in  the  waveguide.  The  signal  polarization  was  oriented  so  that  a  maxi- 
mum reflection  was  obtained  in  the  case  of  the  lower  trace.  In  the 
other  trace,  the  polarization  was  changed  by  90°.  It  is  seen  that  this 
particular  joint  produces  a  stronger  reflection  for  one  polarization  than 
for  the  other.  By  use  of  the  precision  phase  shifter  in  the  receiver  the 
exact  location  of  this  defect  was  found  and  the  particular  joint  that  was 
at  fault  was  sawed  out.  Fig.  7  shows  this  joint  after  the  pipe  had  been 
cut  in  half  through  the  middle.  The  guide  is  quite  smooth  on  the  inside 
in  spite  of  the  discoloration  of  some  solder  that  is  shown  here,  but  on 
the  left-hand  side  of  the  illustration  the  open  crack  is  seen  where  the 
solder  did  not  run  in  properly.  This  causes  the  reflected  pulse  that  shows 
on  the  trace.  The  fact  that  this  crack  is  less  than  a  semi-circumference 
in  length  causes  the  echo  to  be  stronger  for  one  polarization  than  for  the 
other. 


WAVEGUIDE   TESTING    WITH    MILLIMICROSECOND    PULSES 


45 


Fig.  8  shows  the  same  test  for  a  3"  diameter  ahiminum  waveguide 
250  feet  long.  This  line  was  mounted  horizontally  in  the  test  building 
with  compression  couplings  used  at  the  joints.  The  line  expanded  on 
warm  days  hut  the  friction  of  the  mounting  supports  was  so  great  that 
it  pulled  open  at  some  of  the  joints  when  the  temperature  returned  to 
normal.  These  open  joints  produced  reflected  pulses  from  40  to  50  db 
down,  which  are  shown  here.  They  come  at  intervals  equal  to  the  length 
of  one  section  of  pipe,  about  12  feet.  Some  of  these  show  polarization 
effects  where  the  crack  was  more  open  on  one  side  than  on  the  other, 
but  others  are  almost  independent  of  polarization.  These  two  photo- 
graphs of  the  trace  were  taken  with  the  polarization  changed  90°. 

Fig.  9  shows  the  same  test  for  a  3"  diameter  galvanized  iron  wave- 
guide. This  line  had  shown  fairly  high  loss  using  CW  for  measure- 
ments. The  existence  of  a  great  many  echoes  from  random  distances 
indicates  a  rough  interior  finish  in  the  waveguide.  Fig.  10  shows  the 
kind  of  inperfections  in  the  zinc  coating  used  for  galvanizing  which 
caused  these  reflections. 


6.    TESTING  ANTENNA  INSTALLATIONS 

The  use  of  this  equipment  in  testing  waveguide  and  antenna  installa- 
tions for  microwave  radio  repeater  systems  is  shown  in  Fig.  11.  This 
particular  work  was  done  in  cooperation  wdth  A.  B.  Crawford's  antenna 
research  group  at  Holmdel,  who  designed  the  antenna  system.  A  direc- 
tional coupler  was  used  to  observe  energy  reflections  from  the  system 
under  test.  In  this  installation  a  3"  diameter  round  guide  carrying  the 
TEu  mode  was  used  to  feed  the  antenna.  Two  different  waveguide 


TE,,    IN   3"D1AM   aluminum  GUIDE  (250   FT   LONG) 


Fig.  8  —  Reflections  from  several  defective  joints  in  a  dominant  (TEn)  mode 
waveguide.  The  two  trace  photos  are  for  polarizations  differing  by  90  degrees. 


46 


THE    BELL    SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


TO  RECEIVER  ^n 


^— a^IS^rv^ 


i— ^ 


TE 


■^-^Bi 


21 


I 


10 


TE,°  IN   3"  DIAM    GALVANIZED  IRON  GUIDE  (250  FT  LONG) 


Fig.  9  ■ —  Multiple  reflections  from  a  dominant  (TEn)  mode  waveguide  with  a 
rough  inside  surface.  The  two  trace  photos  are  for  polarizations  differing  by  90    I 
degrees. 

joints  are  shown  here.  In  addition,  a  study  was  being  made  of  the  re- 
turn loss  of  the  transition  piece  at  the  throat  of  the  antenna  which  • 
connected  the  3"  waveguide  to  the  square  section  of  the  horn.  The  I 
waveguide  sections  are  about  10  feet  long.  The  overloaded  pulse  at  the 
left  on  the  traces  is  the  leakage  through  the  directional  coupler.  The 


Fig.  10  —  Rough  inside  surface  of  a  galvanized  iron  waveguide  which  produced 
the  reflections  shown  on  Fig.  9. 


I 


WAVEGUIDE   TESTING   WITH   MILLIMICROSECOND    PULSES 


47 


other  echoes  are  associated  with  the  parts  of  the  system  from  which 
they  came  by  the  dashed  Hues  and  arrows  on  the  figure.  A  clamped 
joint  in  the  line  gave  the  reflection  shown  next  following  the  initial 
overloaded  pulse.  A  well  made  threaded  coupling  in  which  the  ends  of 
the  pipe  butted  squarel,y  is  seen  to  have  a  very  much  lower  reflection, 
scarcely  observable  on  this  trace.  Since  there  is  ahvays  reflection  from 
the  mouth  and  upper  reflector  parts  of  this  kind  of  antenna,  it  is  not 
possible  to  measure  a  throat  transition  piece  alone  by  conventional  CW 
methods,  as  the  total  reflected  power  from  the  system  is  measured. 
Here,  use  of  the  resolution  of  this  short  pulse  equipment  completely 
separated  the  reflection  of  the  transition  piece  from  all  other  reflections 
and  made  a  measurement  of  its  performance  possible.  In  this  particular 
case,  the  reflection  from  the  transition  is  more  than  50  db  down  from 
the  incident  signal  which  represents  very  good  design.  As  can  be  seen, 


OPEN   APERTURE 


FIBERGLASS  COVER 
OVER  APERTURE 

REFLECTION    APPEARS 
-^TO   COME   FROM   16  FT 
N    FRONT  OF  HORN    MOUTH 


DIRECTIONAL       TRANSDUCER     CLAMPED      THREADED       ROUND-TO 
COUPLER  JOINT  COUPLING         SQUARE 

TRANSITION 

Fig.  11  —  Waveguide  and  antenna  arrangement  with  trace  photos  showing  re- 
flections from  joints,  transition  section,  and  cover. 


48  THE    BELL   SYSTEM   TECHNICAL  JOURNAL,    JANUARY    1956 

the  reflection  from  the  parabohc  reflector  and  mouth  is  also  finite  low, 
and  this  characterizes  a  good  antenna  installation. 

The  extra  reflected  pulse  on  the  right  of  the  lower  trace  on  Fig.  11 
appeared  when  a  fiberglas  weatherproof  cover  was  installed  over  the 
open  mouth  of  the  horn.  This  cover  by  itself  would  normally  produce  a 
troublesome  reflection.  However,  in  this  antenna,  it  is  a  continuation  of 
one  of  the  side  walls  of  the  horn.  Consequently,  outgoing  signals  strike 
it  at  an  oblique  angle.  Reflected  energy  from  it  is  not  focused  by  the 
parabolic  section  back  at  the  waveguide,  so  the  overall  reflected  power 
in  the  waveguide  was  found  to  be  rather  low.  However,  measuring  it 
with  this  equipment,  we  found  that  an  extra  reflection  appeared  to 
come  from  a  point  16  feet  out  in  front  of  the  mouth  of  the  horn  when  the 
cover  was  in  place.  This  is  accounted  for  by  the  fact  that  energy  re- 
flected obliquely  from  this  cover  bounces  back  and  forth  inside  the 
horn  before  getting  back  into  the  waveguide,  thus  traveling  the  extra 
distance  that  makes  the  measurement  seem  to  show  that  it  comes  from 
16  feet  out  in  front. 

7.    SEPARATION  OF  MODES  ON  A  TIME  BASIS 

If  a  pulse  of  energy  is  introduced  into  a  moderate  length  of  round 
waveguide  to  excite  a  number  of  modes  which  travel  with  different 
group  velocities,  and  then  observed  farther  along  the  line,  or  reflected 
from  a  piston  at  the  end  and  observed  at  the  beginning,  separate  pulses 
will  be  seen  corresponding  to  each  mode  that  is  sent.  This  is  illustrated 


!  t    r 

t     t 

TE„   TMo,TE2, 

TM„         TE3, 

(TEoi) 

^NOT  EXCITED 

TO  RECEIVER 

=^ 

^-^ 

=^^ 

t 

;ft 

t 

TMj, 

TE4I  TE,2 

TM02 

TM3,  AND 

TE5,  TOO 

WEAK  TO 

SHOW 

TE, 


•^^  " 


PROBE  3   DIAM   ROUND  GUIDE 

COUPLING  (WILL  SUPPORT  12  MODES) 


Fig.  12  —  Arrangement  for  showing  mode  separation  on  a  time  basis  in  a  multi- 
mode  waveguide.  The  pulses  in  the  trace  ])]io(o  have  all  traveled  to  the  iiisloii  and 
back.  The  earlier  outgoing  pulse  due  to  direelional  coupler  unbalance  is  not  shown. 


WAVEGUIDE   TESTING   WITH   MILLIMICROSECOND    PULSES 


49 


in  Fig.  12.  In  this  arrangement  energy  was  sent  into  the  round  line  from 
a  probe  inserted  in  the  side  of  the  guide.  This  couples  to  all  of  the  12 
modes  which  can  be  supported,  with  the  exception  of  the  TEoi  circular 
electric  mode.  The  sending  end  of  the  round  guide  was  terminated.  A 
directional  coupler  is  connected  to  the  sending  probe  so  that  the  return 
from  the  piston  at  the  far  end  can  be  observed  on  the  receiver.  Because 
of  the  different  time  that  each  mode  takes  to  travel  one  round  trip  in 
this  waveguide,  which  was  258  feet  long,  separate  pulses  are  seen  for 
each  mode.  The  pulses  in  this  figure  have  been  marked  to  show  which 
mode  is  being  received. 

The  time  of  each  pulse  referred  to  the  outgoing  pulse  was  measured 
and  found  to  check  very  well  with  the  calculated  time.  The  formula  for 
the  time  of  transit  in  the  waveguide  for  any  mode  is: 


T  = 


0.98322V'1  -  VnJ 


[where     T  =  time  in  millimicroseconds 

L  =  length  of  pulse  travel  in  feet 

Vnm    ^^    A /Ac 

X   =  operating  wavelength  in  air 

Ac  =  cutoff  wavelength  of  guide  for  the  mode  involved. 

[  Table  I  —  Calculated  and  Measured  Value  of  Time  for  One 

Round  Trip 


Time  in  Millimicroseconds 

Mode  Designation 

Calculated 

Measured 

1 

TEn 

545 

545 

2 

TMoi 

561 

561 

3 

TE,i 

587 

587 

4 

TMn 

634 

634 

5 

TEoi 

634 

. 

6 

TE31 

665 

665 

7 

TM21 

795 

793 

8 

TE4: 

835 

838 

9 

TE12 

838 



10 

TM„2 

890 

890 

11 

TMn 

1461 

— 

12 

TE51 

1519 

— 

50  THE   BELL   SYSTEM  TECHNICAL   JOURNAL,    JANUARY    1956 

The  calculated  and  measured  \'alue  of  time  for  one  round  trip  is  given 
in  Table  I. 

In  this  experiment  the  operating  wavelength  was  3.35  centimeters 
This  was  obtained  by  measurements  based  on  group  velocit}'  in  a  num- 
ber of  guides  as  well  as  information  about  the  pulse  generator  com- 
ponents. It  represents  an  effective  wa\'elength  giving  correct  time  of 
travel.  The  pulse  occupies  such  a  wide  bandwidth  that  a  measurement 
of  its  wavelength  is  difficult  by  the  usual  means. 

The  dashes  in  the  measured  column  indicate  that  the  mode  was  not 
excited  by  the  probe  or  was  too  weak  to  measure.  These  modes  do  not 
appear  on  the  oscilloscope  trace  photograph. 

The  relative  pulse  heights  can  be  calculated  from  a  knowledge  of  the 
probe  coupling  factors  and  the  line  loss.  The  probe  coupling  factors  as 
given  by  M.  Aronoff  in  unpublished  work  are  expressed  by  the  following 

For  TE„„,  modes: 

P  =  2.390  r—^ 


i 


For  TM„^  modes: 


TV-         L     a    -. 

j\. nm  ^    "flu 


X     X 

P  =  1.195€„  — - 


where 

P  =  ratio  of  probe  coupling  power  in  mode  nm  to  that  in  mode  TEn 

n  =  first  index  of  mode  being  calculated 
Knm  =  Bessel  function  zero  value  for  mode  being  calculated  =  Td/\c 

X  =  wavelength  in  air 

X(,  =  wavelength  in  the  guide  for  the  mode  involved  ' 

Xc  =  cutoff  wavelength  of  guide  for  the  mode  involved 

€„  =  1  for  w  =  0 

€„  =  2  for  n  ?^  0  , 

d  =  waveguide  diameter 

Formulas  for  guide  loss  as  given  by  S.  A.  Schelkunoff  on  page  390  of 
his  book  Elect romagnelir  Waves  for  this  case  where  the  resistivity  of  the 
aluminum  guide  is  4.14  X  10~^  ohms  per  cm  cube  are: 


WAVEGUIDE   TESTING   WITH    MILLIMICROSECOND    PULSES  51 

For  TE„,„  modes: 

a    =    3.805   !    —       2  2    +    V.an     )   (1     "    Vnm) 

\l\n,n      —    n  / 

For  TM„,„  modes: 

a  =  3.805(1  -  VnJy''' 
where: 

a  =  attemiation  of  this  aluminum  guide  in  db 

n  —  first  index  of  mode  being  calculated 
Knm  —  Bessel  function  zero  value  for  mode  being  calculated  =  TrtZ/Xc 

Vnm    =    A/Ac 

X  =  operating  wavelength  in  air 

Xc  =  cutoff  wavelength  of  guide  for  the  mode  involved 

d  =  waveguide  diameter 

Table  II  gives  the  calculated  probe  coupling  factor,  line  loss,  and  rela- 
tive pulse  height  for  each  mode.  In  the  calculation  of  the  latter,  wave 
elUpticity  and  loss  due  to  mode  conversion  were  neglected,  but  the  heat 
loss  given  by  the  preceding  formulas  has  been  increased  20  per  cent  for 
all  modes,  to  take  account  of  surface  roughness.  Relative  pulse  heights 
were  obtained  by  subtracting  the  relative  line  loss  from  twice  the  rela- 
tive probe  coupling  factor.  The  relative  line  loss  is  the  number  in  the 
itable  minus  2.33  db,  the  loss  for  the  TEn  mode. 

The  actual  pulse  heights  on  the  photo  of  the  trace  on  Fig.  12  are  in 
fair  agreement  with  these  calculated  values.  Differences  are  probably 
due  to  polarization  rotation  in  the  guide  (wave  ellipticity)  and  conver- 
sion to  other  modes,  effects  which  were  neglected  in  the  calculations, 
and  which  are  different  for  different  modes. 

Calculated  pulse  heights  with  this  guide  length,  except  for  modes 
near  cutoff,  vary  less  than  the  probe  coupling  factors,  because  line  loss 
is  high  when  tight  probe  coupling  exists.  This  is  to  be  expected,  since 
both  are  the  result  of  high  fields  near  the  guide  walls. 

The  table  of  round  trip  travel  time  shows  that  the  TE41  and  TE12 
modes  are  separated  by  only  three  millimicroseconds  after  the  round 
trip  in  this  waveguide.  They  would  not  be  resolved  as  separate  pulses 
by  this  e(iuipment.  However,  the  table  of  calculated  pulse  heights  shows 
that  the  TE41  pulse  should  be  about  22  db  higher  than  the  TE12  pulse. 


52 


THE    BELL    SYSTEM   TECHNICAL  JOURNAL,    JANUARY    1956 


Table  II  —  Calculated  Probe  Coupling  Factor,  Line  Loss  and 
Pulse  Height  for  Each  Mode 


Mode 

Mode 

Relative  Probe 

1.2  X  Theoretical 

Calculated  Relatix  e 

Number 

Designation 

Coupling  Factor, 
db 

Line  Loss,  db 

Pulse  Heights,  db 

1 

TEu 

0 

2.33 

0 

2 

TMoi 

+0.32 

4.88 

-1.91 

3 

TE2, 

+2.86 

4.85 

+3.20 

4 

TMu 

+2.80 

5.51 

+2.42 

5 

TEo, 

—  00 

1.73 

—  00 

6 

TE31 

+4.82 

8.21 

+3.76 

7 

TM2, 

+  1.82 

6.92 

-0.95 

8 

TE41 

+6.80 

13.86 

+2.07 

9 

TE12 

-8.73 

4.70 

-19.83 

10 

TM02 

-1.68 

7.74 

-8.77 

11 

TMsi 

-0.82 

12.71 

-12.02 

12 

TE51 

+  10.14 

32.09 

-9.48 

Since  the  TE12  pulse  is  so  weak,  it  would  not  show  on  the  trace  even  if 
it  were  resolved  on  a  time  basis.  Coupling  to  the  TM02  mode  is  rather 
weak,  and  the  gain  was  increased  somewhat  at  its  position  on  the  trace 
to  show  its  time  location. 

8.    DELAY  distortion 

Another  effect  of  the  wide  bandwidth  of  the  pulses  used  with  this 
equipment  can  be  observed  in  Fig.  12.  The  pulses  that  have  traveled 
for  a  longer  time  in  the  guide  are  in  the  modes  closer  to  cutoff,  and  are 
on  the  right-hand  side  of  the  oscilloscope  trace.  They  are  broadened 
and  distorted  compared  with  the  ones  on  the  left-hand  side.  This  effect 
is  due  to  delay  distortion  in  the  guide.  This  can  be  explained  by  refer- 
ence to  Fig.  13.  On  this  figure  the  ratio  of  group  velocity  to  the  velocity 
in  an  unbounded  medium  is  shown  plotted  as  a  function  of  frequency 
for  each  of  the  modes  that  can  be  propagated.  The  bandwidth  of  the 
transmitted  pulse  is  indicated  by  the  vertical  shaded  area.  It  will  he 
noticed  that  the  spacing  of  the  pulses  on  the  oscilloscope  trace  on  Fig. 
12  from  left  to  right  in  time  corresponds  to  the  spacing  of  the  group 
velocity  curves  in  the  bandwidth  of  the  pulse  from  top  to  bottom.  De- 
lay distortion  on  these  curves  is  shown  by  the  slope  of  the  line  across 
the  pulse  bandwidth.  If  the  line  were  horizontal,  showing  the  same  group 
velocity  at  all  points  in  the  band,  there  would  be  no  delay  distortion. 
The  greater  the  difference  in  group  A-elocity  at  the  two  edges  of  the 
band,  the  greater  the  delay  distortion.  The  curves  of  Fig.  13  indicate 


WAVEGUIDE   TESTING   WITH   MILLIMICROSECOND    PULSES 


53 


I  that  there  should  be  increasing  amounts  of  delay  distortion  reading 
ifrom  top  to  bottom  for  the  pulse  bandwidth  used  in  these  experiments. 
;The  effect  of  this  delay  distortion  is  to  cause  a  broadening  of  the  pulse. 
Examination  of  the  pulse  pattern  of  Fig.  12  shows  that  the  later  pulses 
corresponding  in  mode  designation  to  the  lower  curves  of  Fig.  13  do  in- 
deed show  a  broadening  due  to  the  increased  delay  distortion.  One 
method  of  reducing  the  effect  of  delay  distortion  is  to  use  frequency 
division  multiplex  so  that  each  signal  uses  a  smaller  bandwidth.  Another 
way,  suggested  by  D.  H.  Ring,  is  to  invert  the  band  in  a  section  of  the 
waveguide  between  one  pair  of  repeaters  compared  with  that  between 
an  adjacent  pair  of  repeaters  so  that  the  slope  is,  in  effect,  placed  in  the 
opposite  direction,  and  delay  distortion  tends  to  cancel  out,  to  a  first 
order  at  least. 

The  (luantitative  magnitude  of  delay  distortion  has  been  expressed 
by  S.  Darlington  in  terms  of  the  modulating  base-band  frequency 
needed  to  generate  two  side  frequencies  which  suffer  a  relative  phase 
error  of  180°  in  traversing  the  line.  This  would  cause  cancellation  of  a 
single  frequency  AM  signal,  and  severe  distortion  using  any  of  the 


1.0 

PULSE    BANDWIDTH 

— >. 

<— 

^^ 

^— 

UJ 

^0.9 

Q. 
</) 

OI 

mo.8 
u. 

z 

^0.7 

1- 

o 

O 

> 
o 

^  0.5 

>- 

o 

3  0.4 

m 

> 

^0.3 

o 

(r 
o 

^0.2 

o 
io., 

0 

■^^H;;^ 

. — 

/^ 

o^^ 

^ 

^ 

^ 

^^ 

/ 

/ 

y 

\a 

X 

y 

/ 

^ 

^ 

/ 

/ 

/ 

6 

^ 

/ 

7 

/ 

f 

// 

< 

f/> 

\ 

^-'^'fA 

y^. 

/ 

/ 

1 

/ 

1 

'L 

f4 

// 

1 

/ 

/ 

1 

1 

/A 

'/ 

// 

// 

L 

^i 

/  1 

7 

\\ 

1 

3  4 

FREQUENCY 


5  6  7  8  9  10 

IN    KILOMEGACYCLES    PER   SECOND 


12 


1  Fig.  1.3  —  Theoretical  group  velocity  vs.  frequency  curves  for  the  3"  diameter 
ivaveguide  used  for  the  tests  shown  on  Fig.  12.  The  vertical  shaded  area  gives  the 
bandwidth  for  the  millimicrosecond  pulses  employed  in  that  arrangement. 


54  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

ordinary  modulation  methods.  Darlington  gives  this  formula: 

^)     ^^^^ 

iLLi/  Vnm 

where : 

jB    =  base  bandwidth  for  180°  out  of  phase  sidebands 

/  =  operating  frequency  (in  same  units  as  jB) 

X  =  wavelength  in  air 

L  =  waveguide  length  (in  same  units  as  X) 

Vnm    =    X/Xe 

Xc  =  cutoff  wavelength  for  the  mode  involved 

With  this  equipment,  the  base  bandwidth  of  the  pulse  is  about  175 
mc,  and  when/5  from  the  formula  above  is  about  equal  to  or  less  than 
this,  pulse  distortion  should  be  observed.  The  following  Table  III  gives 
fB  calculated  from  this  formula  for  the  arrangement  shown  on  Fig.  12. 

It  is  interesting  to  note  that  pulses  in  the  TMu  and  TE31  modes,  for 
which  jB  is  less  than  the  175-mc  pulse  bandwidth,  are  broadened,  but 
not  badly  distorted.  For  the  higher  modes,  where  jB  is  much  less  than 
175  mc,  broadening  and  severe  distortion  are  evident.  Another  example 
is  given  in  the  next  section. 

9.    DELAY   DISTORTION   EQUALIZATION 

If  the  distance  which  a  pulse  travels  in  a  waveguide  is  increased,  its 
delay  distortion  also  increases.  Since  the  group  velocity  at  one  edge  of 
the  band  is  different  than  at  the  other  edge  of  the  band,  the  amount 
by  which  the  two  edges  get  out  of  phase  with  each  other  increases  with 
the  total  length  of  travel,  causing  increased  distortion  and  pulse  broaden- 
ing. The  Darlington  formula  in  the  previous  section  shows  that  jB 
varies  inversely  as  the  square  root  of  the  length  of  travel.  This  efTect 
is  shown  on  Fig.  14.  In  this  arrangement  the  transmitter  was  connected 
to  the  end  of  a  3"  diameter  round  waveguide  107  feet  long  through  a 
small  hole  in  the  end  plate.  A  mode  filter  was  used  so  that  only  the 
TEoi  mode  would  be  transmitted  in  this  Avaveguide.  Through  another 
small  hole  in  the  end  plate  polarized  90°  from  the  first  one,  and  rotated 
90°  around  tlu^  plate,  a  directional  coupler  was  connected  as  shown. 
The  direct  through  guide  of  this  directional  coupler  could  be  short  cir- 
cuited with  a  waveguide  shorting  switch.  Energy  reflected  from  this 


fl 


WAVEGUIDE   TESTING   WITH    MILLIMICROSECOND    PULSES 


55 


Table  III  - 

—  Calculatee 

>  Values  of  fB  foe  the  Arrangement 

Shown  in  Fig.  1 

2 

Mode  Number 

Mode  Designation 

/B  Megacycles 

Remarks 

1 

TEn 

324.0 

2 

TMoi 

237.7 

3 

TEn 

174.9 

4 

TMu 

124.1 

5 

TEoi 

124.1 

Not  excited 

6 

TE31 

105.2 

7 

TMoi 

65.9 

8 

TE41 

59.1 

9 

TEi, 

58.6 

Veiy  weakly  excited 

10 

TMoo 

51.8 

11 

TM3: 

21.3 

Not  observed 

12 

TE51 

20.0 

Not  observed 

NUMBER  OF 

R( 

3UND  TRIPS 

TAPERED 

DELAY 

DISTORTION 

EQUALIZER 


WAVEGUIDE 

SHORTING 

SWITCH 


1/ 


'M 


^ 


te; 


>T0    RECEIVER 


NOT   EQUALIZED 
(SWITCH  CLOSED) 


EQUALIZED 
(SWITCH  OPEN) 


TEqiIN   3    DIAM  ROUND  GUIDE 
(107   FT  LONG) 


Fig.  14  —  The  left-hand  series  of  pulses  shows  the  build  up  of  delay  distortion 
with  increasing  number  of  round  trips  in  a  long  waveguide.  The  right-hand  series 
shows  the  im]irovement  obtained  with  the  tapered  delay  distortion  equalizer 
shown  at  the  right. 


56 


THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


switch  was  then  taken  through  the  directional  coupler  to  the  receiver 
as  shown  by  the  output  arrow.  The  series  of  pulses  at  the  left-hand 
photograph  of  the  oscilloscope  traces  was  taken  with  this  waveguide  i 
shorting  switch  closed.  The  top  pulse  shows  the  direct  leakage  across 
the  inside  of  the  end  plate  before  it  has  traveled  through  the  3"  round 
guide.  The  next  pulse  is  marked  one  round  trip,  having  gone  therefore 
214  feet  in  the  TEoi  mode  in  the  round  waveguide.  The  successive  pulses 
have  traveled  more  round  trips  as  shown  by  the  number  in  the  center 
between  the  two  photographs.  The  effect  of  increased  delay  distortion 
broadening  and  distorting  the  pulse  can  be  seen  as  the  numbers  increase. 
The  values  of  fB  from  the  Darlington  formula  in  the  previous  section 
for  these  lengths  are  given  in  Table  IV. 

It  will  be  noticed  that  pulse  broadening,  and  eventually  severe  dis- 
tortion, occurs  as  fB  decreases  much  below  the  175-mc  pulse  band- 
width. The  effect  is  gradual,  and  not  too  bad  a  pulse  shape  is  seen  until 
fB  is  about  half  the  pulse  bandwidth,  although  broadening  is  very 
evident  earlier. 

When  the  waveguide  short-circuiting  switch  was  opened  so  that  the 
tapered  delay  distortion  equalizer  was  used  to  reflect  the  energy,  in- 
stead of  the  switch,  the  series  of  pulses  at  the  right  was  observed  on 
the  indicator.  It  will  be  noted  that  there  is  much  less  distortion  of  these, 
pulses,  particularly  toward  the  bottom  of  the  series.  The  ones  at  the  top, 
have  less  distortion  than  would  be  expected,  probably  because  of  fre-, 
quency  modulation  of  the  injected  pulse.  The  equalizer  consists  of  a 
long  gradually  tapered  section  of  waveguide  which  has  its  size  reduced 
to  a  point  beyond  cutoff  for  the  frequencies  involved.  Reflection  takes 
place  at  the  point  of  cutoff  in  this  tapered  guide.  For  the  high  frequency 
part  of  the  pulse  bandwidth,  this  point  is  farther  away  from  the  short- 
ing switch  than  for  the  low  frequency  part  of  the  bandwidth.  Conse- 
quently, the  high  frequency  part  of  the  pulse  travels  farther  in  one  round 
trip  into  this  tapered  section  and  back  than  the  low  frequency  part  of 


Table  IV  —  Values  of  fB  from  the  Darlington  Formula 
FOR  the  Arrangement  Show^n  in  Fig.  14 


li 


Round  Trip  Number 

JB  Megacycles 

Round  Trip  Number 

fB  Megacycles 

1 
2 
3 
4 
5 

185.8 

131.4 

107.3 

92.9 

83.1 

6 

7 

8 

9 

10 

75.8 
70.2 
65.7 
61.9 

58.7 

j 

1 

1 

WAVEGUIDE   TESTING   WITH   MILLIMICROSECOND    PULSES  57 

he  pulse.  This  increased  time  of  travel  compensates  for  the  shorter 
ime  of  travel  of  the  high  frequency  edge  of  the  band  in  the  3"  round 
.vaveguide,  so  equalization  takes  place.  Since  this  waveguide  close  to 
cutoff  introduces  considerable  delay  distortion  by  itself,  the  taper  effect 
nust  be  made  larger  in  order  to  secure  the  equalization.  This  can  be 
ilone  by  making  the  taper  sufficiently  gradual.  This  type  of  equalizer 
ntroduces  a  rather  high  loss  in  the  system.  For  this  reason  it  might 
le  used  to  predistort  the  signal  at  an  early  level  in  a  repeater  system, 
ilqualization  by  this  method  was  suggested  by  J.  R.  Pierce. 

.0.    MEASURING  MODE  CONVERSION  FROM  ISOLATED  SOURCES 

I  One  of  the  important  uses  of  this  equipment  has  been  for  the  meas- 
irement  of  mode  conversion.  W.  D.  Warters  has  cooperated  in  develop- 
ng  techniques  and  carrying  out  such  measurements.  One  of  the  prob- 
ems  in  the  design  of  mode  filters  used  for  suppressing  all  modes  except 
;he  circular  electric  ones  in  round  multimode  guides  is  mode  conversion. 
Since  these  mode  filters  have  circular  symmetry,  conversion  can  take 
alace  only  to  circular  electric  modes  of  order  higher  than  the  TEoi  mode. 
This  conversion  is,  however,  a  troublesome  one,  since  these  higher 
Drder  circular  modes  cannot  be  suppressed  by  the  usual  type  of  filter. 

An  arrangement  for  measuring  mode  conversion  at  such  mode  filters 
rom  the  TEoi  to  the  TE02  mode  is  being  used  with  the  short  pulse  equip- 
:nent.  This  employs  a  400-foot  long  section  of  the  b"  diameter  line.  Be- 
ause  the  coupled- line  transducer  available  had  too  high  a  loss  to  TE02 ,  a 
3ombined  TEoi  —  TE02  transducer  was  assembled.  It  uses  one-half  of 
:he  round  waveguide  to  couple  to  each  mode.  Fig.  15  shows  this  device. 

The  use  of  this  transducer  and  line  is  illustrated  in  Fig.  16.  Pulses  in 
:he  TEoi  mode  are  sent  into  the  waveguide  by  the  upper  section  of  the 
transducer  as  shown.  Some  of  the  TEoi  energy  goes  directly  across  to 
ohe  TE02  transducer  and  appears  as  the  outgoing  pulse  with  a  level 
down  about  32  db.  This  is  useful  as  a  time  reference  in  the  system  and 
s  shown  as  the  outgoing  pulse  in  the  photo  of  the  oscilloscope  trace 
ibove.  The  main  energy  in  the  TEoi  mode  propagates  down  the  line  as 
hown  by  dashed  line  2,  which  is  the  path  of  this  wave.  Most  of 
ohis  energy  goes  all  the  way  to  the  reflecting  piston  at  the  far  end  and 
ohen  returns  to  the  TE02  transducer  where  it  gives  a  pulse  which  is 
narked  TEoi  round  trip  on  the  trace  photograph  above.  Two  thirds  of 
;he  way  from  the  sending  end  to  the  piston,  the  mode  filter  being  meas- 
ired  is  inserted  in  the  line.  When  the  TEoi  mode  energy  comes  to  this 
node  filter,  a  small  amount  of  it  is  converted  to  the  TE02  mode.  This 


58 


THE    BELL   SYSTEM   TECHNICAL   JOURNAL 


Fig.  15  —  A  special  experimental  transducer  for  injecting  the  TEoi  mode  and' 
receiving  the  converted  TE02  mode  in  a  5"  diameter  waveguide. 


continues  to  the  piston  by  path  4  (with  dashed  Hnes  and  crosses) 
and  then  returns  and  is  received  by  the  TE02  part  of  the  transducer. 
This  appears  on  the  trace  photo  as  the  TE02  first  conversion.  When  the 
main  TEoi  energy  reflected  by  the  piston  comes  back  to  the  mode  filter, 
conversion  again  takes  place  to  TE02  •  This  is  shown  by  path  3  hav- 
ing dashed  lines  and  circles.  This  returns  to  the  TE02  part  of  the  trans- 
ducer and  appears  on  the  trace  photo  as  the  TE02  second  conversion. 
In  addition,  a  small  amount  of  energy  in  the  TE02  mode  is  generated 
by  the  TEoi  upper  part  of  the  transducer.  It  is  shown  by  path  5,  having' 


OUTGOING  PULSE 


TEoi 

ROUND 

TRIP 


TE02 

SECOND 

CONVERSION 


TE02 

FIRST 

CONVERSION 


TE02 

ROUND 

TRIP 


' 


MODE   FILTER 


Fig.  16  —  Trace  photos  and  waveguide  paths  traveled  when  measuring  TEoi, 
to  TE02  mode  conversion  at  a  mode  filter  with  the  transducer  shown  on  Fig.  15 


All 


WAVEGUIDE   TESTING   WITH   MILLIMICROSECOND    PULSES  59 

jihort  dashes.  This  goes  down  through  the  waveguide  to  the  far  end 
Ijiston  and  back,  and  is  received  by  the  TE02  transducer  and  shown  as 
[he  pulse  marked  TE02  round  trip.  The  pulse  marked  TEoi  round  trip 
las  a  time  separation  from  the  outgoing  pulse  which  is  determined  by 
,he  group  velocity  of  TEoi  waves  going  one  round  trip  in  the  guide.  The 
|rEo2  round  trip  pulse  appears  at  a  time  corresponding  to  the  group 
/elocity  of  the  TE02  mode  going  one  round  trip  in  the  guide.  Spacing  the 
node  filter  two-thirds  of  the  way  down  produces  the  two  conversion 
:)ulses  equally  spaced  between  these  two  as  shown  in  Fig.  16.  The  first 
ponversion  pulse  appears  at  a  time  which  is  the  sum  of  the  time  taken 
or  the  TEoi  to  go  down  to  the  filter  and  the  TE02  to  go  from  the  filter 
uo  the  piston  and  back  to  the  receiver.  Because  of  the  slower  velocity 
bf  the  TE02 ,  this  appears  at  the  time  shown,  since  it  was  in  the  TE02 
node  for  a  longer  time  than  it  was  in  the  TEoi  mode.  The  second  con- 
[/ersion,  which  happened  when  TEoi  came  back  to  the  mode  filter,  comes 
jiarlier  in  time  than  the  first  conversion,  since  the  path  for  this  signal 
ivas  in  the  TEoi  mode  longer  than  it  was  in  the  TE02  mode.  This  arrange- 
,nent  gives  very  good  time  separation,  and  makes  possible  a  measure- 
Inent  of  the  amount  of  mode  conversion  taking  place  in  the  mode  filters, 
viode  conversion  from  TEoi  to  TE02  as  low  as  50  to  55  db  down,  can  be 
neasured  with  this  equipment. 

Randomly  spaced  single  discontinuities  in  long  waveguides  can  be 
ocated  by  this  technique  if  they  are  separated  far  enough  to  give  in- 
lividually  resolved  short  pulses  in  the  converted  mode.  Fig.  17  shows 


CONVERSION 

FIRST    CONVERSION             AT   FAR  END  SECOND  CONVERSION 

AT    NEAR   END                        SQUEEZED  AT    NEAR  END 

SQUEEZED  SECTION                   SECTION  SQUEEZED  SECTION 


TO  RECEIVER 


TEJo     — *-       TEq,  TE2,    -• »-    TE,o  NEAR  END  250  FT  OF  FAR  END 

TRANSDUCER  COUPLED  LINE  SQUEEZED       3"DIAM  ROUND      SQUEEZED 

TRANSDUCER  SECTION  GUIDE  SECTION 

Fig.  17  —  Arrangement  used  to  explain  the  measurement  and  location  of  mode 
onversion  from  isolated  sources.  A  deliberately  squeezed  section  was  placed 
t  each  end  of  the  long  waveguide,  producing  the  pulses  shown  in  the  trace  photo. 


60  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

an  arrangement  having  oval  sections  deliberately  placed  in  the  wave- ' 
guide  in  order  to  explain  the  method.  Pure  TEoi  excitation  is  vised,  and 
the  converted  TE21  mode  observed  with  a  coupled  line  transducer  giv- ; 
ing  an  output  for  that  mode  alone.  ; 

Let  us  consider  first  what  would  happen  with  the  far-end  squeezed; 
section  alone,  omitting  the  near-end  squeezed  section  from  considera-  • 
tion.  The  injected  TEoi  mode  signal  would  then  travel  down  the  250 , 
feet  of  3"  diameter  round  waveguide  to  the  far  end  with  substantially, 
no  mode  conversion  at  the  level  being  measured.  At  this  point  it  goes 
through  the  squeezed  section.  Conversion  now  takes  place  from  the  TEou, 
mode  to  the  TE21  mode.  Both  these  modes  after  reflection  from  the  piston 
travel  back  up  the  waveguide  to  the  sending  end.  The  group  velocity 
of  the  TE21  mode  is  higher  than  the  group  velocity  of  the  TEoi  mode,  so 
energy  in  these  two  modes  separates,  and  if  a  coupling  system  were 
used  to  receive  energy  in  both  modes,  two  pulses  would  appear,  with  at 
time  separation  between  them.  In  this  case,  since  the  receiver  is  con- 
nected to  the  line  through  the  coupled  line  transducer  which  is  responsive 
only  to  the  TE21  mode,  only  one  pulse  is  seen,  that  due  to  this  mode 
alone.  This  is  the  center  pulse  in  the  trace  photograph  at  the  top  of 
Fig.  17.  If  only  one  mode  conversion  point  at  the  far  end  of  the  guide 
exists,  only  this  one  pulse  is  seen  at  the  receiver.  It  would  be  spaced  a 
distance  away  from  the  injected  outgoing  pulse  that  corresponds  m:^ 
time  to  one  trip  of  the  TEoi  mode  down  to  the  far  end  and  one  trip  of  || 
the  TE21  mode  from  the  far  end  back  to  the  receiver. 

Now  let  us  consider  what  would  happen  if  the  near-end  squeezed  sec- 
tion alone  were  present.  When  the  TEqi  wave  passes  the  oval  section! 
just  beyond  the  coupled  line  transducer,  conversion  takes  place,  andi 
the  energy  travels  down  the  line  in  both  the  TEoi  and  the  TE21  modes,:; 
at  a  higher  group  velocity  in  the  TE21  mode.  These  two  signals  are  re- 
flected by  the  piston  at  the  far  end  and  return  to  the  sending  end.  The 
TE21  signal  comes  through  the  coupled  line  transducer  and  appears  as 
the  pulse  at  the  left  of  the  photo  shown  on  Fig.  17.  Now  the  TEoi  energy 
has  lagged  behind  the  TE21  energy,  and  when  it  gets  back  to  the  near- 
end  squeezed  section,  a  second  mode  conversion  takes  place,  and  TE21 
mode  energy  is  produced  which  comes  through  the  coupled  line  trans-: 
ducer  and  appears  at  the  receiver  at  the  time  of  the  right  hand  pulse. 
The  spacing  between  these  two  pulses  is  equal  to  the  difference  in  round 
trip  times  between  the  two  modes. 

In  general,  for  a  single  conversion  source  occurring  at  any  point  in 
the  line,  two  pulses  will  appear  on  the  scope.  The  spacing  between  these 
pulses  corresponds  to  the  difference  in  group  velocity  between  the  modes. 


WAVEGUIDE   TESTING   AVITH   MILLIMICROSECOND   PULSES  61 

{from  the  point  of  the  discontiimity  down  to  the  piston  at  the  far  end, 

land  then  back  to  the  discontinuity.  If  the  discontinuity  is  at  the  far 

lend,  this  time  difference  becomes  zero,  and  a  single  pulse  is  seen.  By 

i  [making  a  measurement  of  the  pulse  spacing,  the  location  of  a  single 

i  icon  version  point  can  be  determined. 

[      In  the  arrangement  illustrated  in  Fig.  17,  two  isolated  sources  of 

j  conversion  existed.  They  were  spaced  far  enough  apart  so  that  they 

\  were  resolved  by  this  equipment,  and  all  three  pulses  were  observed. 

The  two  outside  pulses  were  due  to  the  first  conversion  point.  The  center 

pulse  was  caused  by  the  other  squeeze,  which  was  right  at  the  reflecting 

|:)iston.  If  this  conversion  point  had  been  located  back  some  distance 

rom  the  piston,  it  would  have  produced  two  conversion  pulses  whose 

'spacing  could  be  used  to  determine  the  location  of  the  conversion  point. 

I    The  coupled-line  transducers  are  calibrated  for  coupling  loss  by  send- 

ng  the  pulse  through  a  directional  coupler  into  the  branch  normally 

ised  for  the  output  to  the  receiver.  This  gives  a  return  loss  from  the 

lirectional  coupler  equal  to  twice  the  transducer  loss  plus  the  round 

rip  line  loss. 

1.    MEASURING    DISTRIBUTED    MODE    CONVERSION    IN    LONG    WAVEGUIDES 

;  Measurements  of  mode  conversion  from  TEoi  to  a  number  of  other 
nodes  have  been  made  with  5"  diameter  guides  using  this  equipment, 
rhe  arrangement  of  Fig.  18  was  set  up  for  this  purpose.  This  is  the  same 
IS  Fig.  17,  except  that  a  long  taper  was  used  at  the  input  end  of  the  5" 
waveguide,  and  a  movable  piston  installed  at  the  remote  end. 

One  of  the  converted  modes  studied  with  this  apparatus  arrange- 
uent  was  the  TMu  mode,  which  is  produced  by  bends  in  the  guide, 
rhis  mode  has  the  same  velocity  in  the  waveguide  as  the  TEoi  mode. 
Therefore  energy  components  converted  at  different  points  in  the  line 
tay  in  phase  with  the  injected  TEoi  mode  from  which  they  are  converted, 
rhere  is  never  any  time  separation  between  these  modes,  and  a  single 

TO    RECEIVER 


■  I  ■■'■    '■■■  ^ 


^^i 


Si 


TEro— *TE^,  COUPLED    LINE  -^^p^P,  5„  0,^^^  MOVABLE 

TRANSDUCER  TRANSDUCER  HOLMDEL    LINE  PISTON 

FOR   THE   MODE 
BEING  MEASURED 

Fig.  18  —  Arrangement  used  for  measuring  mode  conversion  in  the  5"  diameter 
aveguides  at  Holmdel. 


62 


THE   BELL    SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


narrow  pulse  like  the  transmitted  one  is  all  that  appears  on  the  indicator 
oscilloscope.  It  is  not  possible  from  this  to  get  any  information  about 
the  location  or  extent  of  the  conversion  points  in  the  line.  Moving  the 
far  end  piston  does  not  change  the  relative  phases  of  the  modes,  so  no 
changes  are  seen  in  indicator  pattern  or  pulse  level  as  the  piston  is 
moved.  For  the  Holmdel  waveguides,  which  are  about  500  feet  long, 
the  total  round  trip  T]\In  mode  converted  level  varies  from  32  to  36  db 
below  the  input  TEoi  mode  level  over  a  frequency  range  from  8,800  to 
9,600  mc  per  second. 

All  the  other  modes  have  velocities  that  are  different  than  that  of 
the  TEoi  mode.  ^Vhen  mode  conversion  takes  place  at  many  closely 
spaced  points  along  the  waveguide,  the  pulses  from  the  various  sources 
overlap,  and  phasing  effects  take  place.  In  general,  a  filled-in  pulse 
much  longer  than  the  injected  one  is  observed.  The  maximum  possible, 
but  not  necessary,  pulse  length  is  equal  to  the  difference  in  time  re- 
quired for  the  TEoi  mode  and  the  converted  mode  to  travel  the  total 
waveguide  length  being  observed.  The  phasing  effects  within  the  broad- 
ened pulse  change  its  height  and  shape  as  a  function  of  frequency  and 
line  length. 

Measurements  of  mode  conversion  from  TEoi  to  TE31  in  these  wave- 
guides illustrate  distributed  sources  and  piston  phasing  effects.  The 
TE3,  mode  has  a  group  velocity  1.4  per  cent  slower  than  the  TEoi  mode. 
For  a  full  round  trip  in  the  500-foot  lines,  assuming  conversion  at  the 
imput  end,  this  causes  a  time  separation  of  about  two  and  one  half 
pulse  widths  between  these  two  modes.  The  received  pulse  is  about  two 
and  a  half  times  as  long  as  the  injected  pulse,  indicating  rather  closely 
spaced  sources  over  the  whole  line  length.  For  one  far-end  piston  posi- 
tion, the  received  pattern  is  shown  as  the  upper  trace  in  Fig.  19.  As 
the  piston  is  moved,  the  center  depressed  part  of  the  trace  gradually 


ImK.  10  —  Hocoivcd  pulsr  patterns  willi  llic  .irraiijicnuMit  of  Fig.  IS  used  for 
studying  conversion  to  tlie  Tlvn  mode. 


WAVEGUIDE   TESTING   WITH    MILLIMICROSECOND    PULSES  63 

rises  until  the  pattern  shown  in  the  lower  trace  is  seen.  As  the  piston 
is  moved  farther  in  the  same  direction  the  trace  gradually  changes  to 
have  the  appearance  of  the  upper  photo  again.  Moving  the  far-end 
piston  changes  the  phase  of  energy  on  the  return  trip,  and  thus  it  can 
be  made  to  add  to,  or  nearly  cancel  out,  conversion  components  that 
originated  ahead  of  the  piston.  When  the  time  separation  becomes 
great  enough  to  prevent  overlapping  in  the  pulse  ^^^dth,  phasing  effects 
cannot  take  place,  therefore,  the  beginning  and  end  of  the  spread-out 
received  pulse  are  not  affected  by  moving  the  piston.  Energy  converted 
at  the  sending  end  of  the  guide  travels  the  full  round  trip  to  the  piston 
and  back  in  the  slower  TE31  mode,  and  thus  appears  at  the  latest  time, 
which  is  at  the  right-hand  end  of  the  received  pulse.  Conversion  at  the 
piston  end  returns  at  the  center  of  the  pulse,  and  conversion  on  the 
return  trip  comes  at  earlier  times,  at  the  left-hand  part  of  the  pulse. 
The  TEoi  mode  has  less  loss  in  the  guide  than  the  TE31  mode.  Since  the 
energy  in  the  earlier  part  of  the  received  pulse  spent  a  greater  part  of 
the  trip  in  the  lower  loss  TEoi  mode  before  conversion,  the  output  is 
higher  here,  and  slopes  off  toward  the  right,  where  the  later  returning 
energy  has  gone  for  a  longer  distance  in  the  higher  loss  mode.  The  pulse 
height  at  the  maximum  shows  the  converted  energy  from  that  part  of 
the  line  to  be  between  30  and  35  db  below  the  incident  TEoi  energy 
level  over  the  measured  band\\ddth. 

Measurements  of  mode  conversion  from  TEoi  to  TE21  in  these  wave- 
guides show  these  same  effects,  and  also  a  phasing  effect  as  a  function 
of  frequency.  The  TE21  mode  has  a  group  velocity  2.4  per  cent  faster 
than  the  TEoi  mode.  For  a  full  round  trip  in  the  guides,  this  is  a  time 
separation  of  about  four  pulse  mdths  between  the  modes.  At  one  fre- 
quency and  one  far-end  piston  position,  the  TE21  response  shown  as  the 
top  trace  of  Fig.  20  was  obtained.  Moving  the  far-end  piston  gradually 
changed  this  to  the  second  trace  from  the  top,  and  further  piston  mo- 
tion changed  it  back  again.  This  is  the  same  kind  of  piston  phasing  effect 
observed  in  the  TE31  mode  conversion  studies.  The  irregular  top  of  this 
broadened  pulse  indicates  fewer  conversion  points  than  for  the  TE31 
mode,  or  phasing  effects  along  the  guide  length.  Since  the  TE21  mode 
has  a  higher  group  velocity'  than  the  TEoi  mode,  energy  converted  at 
the  beginning  of  the  guide  returns  at  the  earlier  or  left-hand  part  of  the 
pulse,  and  conversions  on  the  return  trip,  having  traveled  longer  in 
the  slower  TEoi  mode,  are  on  the  right-hand  side  of  the  pulse.  This  is 
just  the  reverse  of  the  situation  for  the  TE31  mode.  Since  the  loss  in  the 
TE21  mode  is  higher  than  in  the  TEoi  mode,  the  right  side  of  this  broad- 
ened pulse  is  higher  than  the  left  side,  as  the  energy  in  the  left  side  has 


64 


THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


gone  further  in  the  higher  loss  TE21  mode.  Conversions  from  the  piston 
end  of  the  guide  return  in  the  center  of  the  pulse,  and  only  in  this  re- 
gion do  piston  phasing  effects  appear.  As  the  frequency  is  changed  the  ' 
pattern  changes,  until  it  reaches  the  extreme  shape  shown  in  the  next- 
to-the-bottom  trace,  with  this  narrower  pulse  coming  at  a  time  corre- 
sponding to  the  center  of  the  broadened  pulse  at  the  top.  Further  fre- 
quency change  in  the  same  direction  returns  the  shape  to  that  of  the 
top  traces.  At  the  frequency  giving  the  received  pulse  shown  on  the 
next-to-the-bottom  trace,  moving  the  far-end  piston  causes  a  gradual 
change  to  the  shape  shown  on  the  lowest  trace.  This  makes  it  appear 
as  if  the  mode  conversion  were  coming  almost  entirely  from  the  part  of 
the  guide  near  the  piston  end  at  this  frequency.  The  upper  traces  appear 
to  show  that  more  energy  is  converted  at  the  transducer  end  of  the 
waveguide  at  that  frequency.  It  would  seem  that  at  certain  frequencies 
some  phase  cancellation  is  taking  place  between  conversion  points 
spaced  closely  enough  to  overlap  within  the  pulse  width .  At  frequencies 
between  the  ones  giving  traces  like  this,  the  appearance  is  more  like 
that  shown  for  the  TE31  mode  on  Fig.  19  except  for  the  slope  across  the 
top  of  the  pulse  being  reversed.  The  highest  part  of  this  TEoi  pulse  is 


Fiff.  20  —  Received  pulse  patterns  witli  the  urrangemeiit  of  Fig.  18  used  for 
studying  conversion  to  the  TE21  mode. 


WAVEGUIDE   TESTING   WITH    MILLIMICKOSECOND    PULSES  65 

24  to  27  db  below  the  injected  TEoi  pulse  level  for  the  5"  diameter 
Holmdel  waveguides. 

12.    CONCLUDING  REMARKS 

The  high  resolution  obtainable  with  this  millimicrosecond  pulse 
equipment  provides  information  difficult  to  obtain  by  any  other  means. 
These  examples  of  its  use  in  waveguide  investigations  indicate  the 
possibilities  of  the  method  in  research,  design  and  testing  procedures. 
It  is  being  used  for  many  other  similar  purposes  in  addition  to  the  illus- 
tratio)is  given  here,  and  no  doubt  many  more  uses  will  be  found  for 
such  short  pulses  in  the  future. 

REFERENCES 

1.  S.  E.  Miller  and  A.  C.  Beck,  Low-loss  Waveguide  Transmission,  Proc.  I.R.E., 

41,  pp.  348-358,  March,  1953. 

2.  S.  E.  Miller,  Waveguide  As  a  Communication  Medium,  B.  S.  T.  J.,  33,  pp.  1209- 

1265,  Nov.,  1954. 

3.  C.  C.  Cutler,  The  Regenerative  Pulse  Generator,  Proc.  I.R.E.,  43,  pp.  140- 

148,  Feb.,  1955. 

4.  S.  E.  Miller,  Coupled  WaveTheory  and  Waveguide  Applications,  B.  S.  T.  J.,  33, 

pp.  661-719,  May,  1954. 


Experiments  on  the  Regeneration  of 
Binary  Microwave  Pulses 

By  O.  E.  DeLANGE 

(Manuscript  received  September  7,  1955) 

A  sifnple  device  has  been  produced  for  regenerating  binary  pulses  directly 
at  microwave  frequencies.  To  determine  the  capabilities  of  such  devices  one 
of  them  was  included  in  a  circidating  test  loop  in  which  pidse  groups  were 
passed  through  the  device  a  large  number  of  titnes.  Residts  indicate  that 
even  in  the  presence  of  serious  noise  and  bandwidth  limitations  pidses  can 
be  regenerated  many  times  and  still  shotv  no  noticeable  deterioration.  Pic- 
tures of  circulated  pidses  are  included  which  illustrate  performance  of  the 
regenerator. 

INTRODUCTION 

The  chief  advantage  of  a  transmission  system  employing  Ijinary  pulses 
resides  in  the  possibility  of  regenerating  such  pulses  at  intervals  along 
the  route  of  transmission  to  prevent  the  accumulation  of  distortion  due 
to  noise,  bandwidth  limitations  and  other  effects.  This  makes  it  possible 
to  take  the  total  allowable  deterioration  of  signal  in  each  section  of  a 
long  relay  system  rather  than  having  to  make  each  link  sufficiently  good 
to  prevent  total  accumulated  distortion  from  becoming  excessive.  This 
has  been  pointed  out  by  a  number  of  writers. i-- 

W.  M.  GoodalP  has  shown  the  feasibility  of  transmitting  television 
signals  in  binary  form.  Such  transmission  reciuires  a  considerable  amount 
of  bandwidth;  a  seven  digit  system,  for  example,  would  require  trans- 
mission of  seventy  million  pulses  per  second.  This  need  for  wide  bands 
makes  the  microwave  range  an  attractive  one  in  which  to  work.  S.  E. 
Miller*  has  pointed  out  that  a  binary  system  employing  regeneration 
might  prove  to  be  especially  advantageous  in  waveguide  transmission. 

1  B.  M.  Oliver,  J.  R.  Pierce  and,  C.  E.  Shannon,  The  Pliilosophv  of  PCM,  Proc. 
I.  R.E.,  Nov.,  1948. 

'^  L.  A.  Meacham  and  Iv  Peterson,  An  Experimental  Multichannel  Pulse  Code 
Modulation  System  of  Toll  Quality,  B.  S.  T.  J.,  Jan.  1948. 

'  W.  M.  Goodall,  Television  l)y  Pulse  Code  Modulation,  B.  S.  T.  J.,  Jan.,  1951. 

*  S.  E.  Miller,  Waveguide  as  a  Communication  Medium,  B.  S.  T.  J.,  Nov.,  1954. 

67 


68  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


INPUT 


FILTER 


AUTOMATIC 

GAIN 

CONTROL 


REGENERATOR 


DETECTOR 


TIMING 

WAVE 

GENERATOR 


FILTER 


OUTPUT 


Fig.  1  —  A  typical  regenerative  repeater  shown  in  block  form. 


That  the  Bell  System  is  interested  in  the  long-distance  transmission 
of  television  and  other  broad-band  signals  is  evident  from  the  number 
of  miles  of  such  broad-band  circuits,  both  coaxial  cable  and  microwave 
radio, ^  now  in  service.  These  circuits  provide  high-grade  transmission 
because  each  repeater  was  designed  to  have  a  very  fiat  frequency  charac- 
teristic and  linear  phase  over  a  considerable  bandwidth.  Furthermore, 
these  characteristics  are  very  carefully  maintained.  For  a  binary  pulse 
system  employing  regeneration  the  requirements  on  flatness  of  band  and 
linearity  of  phase  can  be  relaxed  to  a  considerable  degree.  The  compo- 
nents for  such  a  system  should,  therefore,  be  simpler  and  less  expensive 
to  build  and  maintain.  Reduced  maintenance  costs  might  well  prove  to 
be  the  chief  virtue  of  the  binary  system. 

Since  the  chief  advantage  of  a  binary  system  lies  in  the  possibility  of 
regeneration  it  is  obvious  that  a  very  important  part  of  such  a  system  is 
the  regenerative  repeater  employed.  Fig.  1  shows  in  block  form  a  typical 
broad-band,  microwave  repeater.  Here  the  input,  which  might  come  from 
either  a  radio  antenna  or  from  a  waveguide,  is  first  passed  through  a 
proper  microwave  filter  then  amplified,  probably  by  a  traveling-wave 
amplifier.  The  amplified  pulses  of  energy  are  regenerated,  filtered,  am- 
plified and  sent  on  to  the  next  repeater.  The  experiment  to  be  described 
here  deals  primarily  with  the  block  labeled  "Regenerator"  on  Fig.  1. 

In  these  first  experiments  one  of  our  main  objectives  was  to  keep  the 
repeater  as  simple  as  possible.  This  suggests  regeneration  of  pulses 
directly  at  microwave  frequency,  which  for  this  experiment  was  chosen 
to  be  4  kmc.  It  was  suggested  by  J.  R.  Pierce  and  W.  D.  Lewis,  both  of 
Bell  Telephone  Laboratories,  that  further  simplification  might  be  made 
possible  by  accepting  only  partial  instead  of  complete  regeneration. 
This  suggestion  was  adopted. 

For  the  case  of  complete  regeneration  each  incoming  pulse  inaugurates 
a  new  pulse,  perfect  in  shape  and  correctly  timed  to  be  sent  on  to  the 

'A.  A.  Roetken,  K.  D.  Smith  and  R.  W.  Friis,  The  TD-2  System,  B.  S.  T.  J., 
Oct.,  1951,  Part  II. 


REGENERATION    OF    BINARY   MICROWAVE    PULSES  69 

next  repeater.  Thus  noise  and  other  disturbing  effects  are  completely 
eliminated  and  the  output  of  each  repeater  is  identical  to  the  original 
signal  which  entered  the  system.  For  the  case  of  partial  regeneration 
incoming  pulses  are  retimed  and  reshaped  only  as  well  as  is  possible  with 
simple  equipment.  Obviously  the  difference  between  complete  and  partial 
.  regeneration  is  one  of  degree. 

One  object  of  the  experiment  was  to  determine  how  well  such  a  partial 
regenerator  would  function  and  what  price  must  be  paid  for  employing 
partial  instead  of  complete  regeneration.  The  regenerator  developed 
consists  simply  of  a  waveguide  hybrid  junction  with  a  silicon  crystal 
diode  in  each  side  arm.  It  appears  to  meet  the  requirement  of  simplicity 
in  that  it  combines  the  functions  of  amplitude  slicing  and  pulse  retiming 
in  one  unit.  A  detailed  description  of  this  unit  will  be  given  later.  Al- 
though the  purpose  of  this  experiment  was  to  determine  what  could  be 
accomplished  in  a  very  simple  repeater  we  must  keep  in  mind  that 
superior  performance  would  be  obtained  from  a  regenerator  which  ap- 
proached more  nearly  the  ideal.  For  some  applications  the  better  re- 
generator might  result  in  a  more  economical  system  even  though  the 
regenerator  itself  might  be  more  complicated  and  more  expensive  to 
produce. 

METHOD    OF   TESTING 

The  regeneration  of  pulses  consists  of  two  functions.  The  first  function 

is  that  of  removing  amplitude  distortions,  the  second  is  that  of  restoring 

each  pulse  to  its  proper  time.  The  retiming  problem  divides  into  two 

[parts  the  first  of  which  is  the  actual  retiming  process  and  the  second 

!  that  of  obtaining  the  proper  timing  pulses  with  which  to  perform  this 

lifunction.  In  a  practical  commercial  system  timing  information  at  a 

[repeater  would  probably  be  derived  from  the  incoming  signal  pulses. 

There  are  a  number  of  problems  involved  in  this  recovery  of  timing 

pulses.  These  are  being  studied  at  the  present  time  but  were  avoided  in 

the  experiment  described  here  by  deriving  such  information  from  the 

local  synchronizing  gear. 

Since  the  device  we  are  dealing  with  only  partially  regenerates  pulses 
it  is  not  enough  to  study  the  performance  of  a  single  unit  —  we  should 
•like  to  have  a  large  number  operating  in  tandem  so  that  we  can  observe 
'what  happens  to  pulses  as  they  pass  through  one  after  another  of  these 
Tegenerators.  To  avoid  the  necessity  of  building  a  large  number  of  units 
the  pulse  circulating  technique  of  simulating  a  chain  of  repeaters  was 
j  employed.  Fig.  2  shows  this  circulating  loop  in  block  form. 


70 


THE    BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 


HYBRID 

JUNCTION 

'  NO.  3 


CW 

OSCILLATOR 

(4  KMC) 


TRAVELING   WAVE 

AMPLIFIER 

(NOISE   GENERATOR) 


Fig.  2  —  The  circulating  loop. 


To  provide  RF  test  pulses  for  this  loop  the  output  of  a  4  kmc,  cw 
oscillator  is  gated  by  baseband  pulse  groups  in  a  microwave  gate  or 
modulator.  The  resultant  microwa\-e  pulses  are  fed  into  the  loop  (heavy 
line)  through  hybrid  junction  No.  1.  They  are  then  amplified  by  a  trav- 
eling-wave amplifier  the  output  of  which  is  coupled  to  the  pulse  regen- 
erator through  another  hybrid  junction  (No.  2).  The  purpose  of  this 
hybrid  is  to  provide  a  position  for  monitoring  the  input  to  the  regen- 
erator. A  monitoring  position  at  the  output  of  the  regenerator  is  pro- 
vided by  a  third  hybrid,  the  main  output  of  which  feeds  a  considerable 
length  of  waveguide  which  provides  the  necessary  loop  delay.  At  the  far 
end  of  the  waveguide  another  hybrid  (No.  4)  makes  it  possible  to  feed 
noise,  which  is  derived  from  a  traveling-wave  amplifier,  into  the  loop. 
The  combined  output  after  passing  through  a  band  pass  filter  is  ampli- 


REGENEKATION    OF   BINARY   MICROWAVE    PULSES 


71 


fied  by  another  traveling-wave  amplifier  and  fed  back  into  the  loop  in- 
put thus  completing  the  circuit. 

The  synchronizing  equipment  starts  out  with  an  oscillator  going  at 
approximately  78  kc.  A  pulse  generator  is  locked  in  step  with  this  os- 
cillator. The  output  of  the  pulser  is  a  negative  3  microsecond  pulse  as 
shown  in  Fig.  3A.  After  being  amplified  to  a  level  of  about  75  volts 
this  pulse  is  applied  to  the  helix  of  the  first  traveling-wave  tube  to  re- 
I  duce  the  gain  of  this  tube  during  the  3-microsecond  interval.  Out  of  each 
12.8/xsec  interval  pulses  are  allowed  to  circulate  for  O.S/xsec  but  are  blocked 
I  for  the  remaining  3Msec  thus  allowing  the  loop  to  return  to  the  quiescent 
i  condition  once  during  each  period  as  shown  on  Figs.  3A  and  3C. 

The  S^sec  pulse  also  synchronizes  a  short-pulse  generator.  This  unit 
delivers  pulses  which  are  about  25  millimicroseconds  long  at  the  base 
and  spaced  by  12.8/isec,  i.e.,  Avith  a  repetition  frequency  of  78  kc.  See 
Fig.  3B. 

In  order  to  simulate  a  PCM  system  it  was  decided  to  circulate  pulse 


CIRCULATING   INTERVAL 
9.8/ZS 


QUENCHING 
INTERVAL 

-3//S-*| 


(A)  GATING  CYCLE 


(B)  SHORT  SYNCHRONIZING  PULSES 


--24  GROUPS  OF  PULSES 


(C)  CIRCULATING  PULSE  GROUPS 


GROUP     GROUP     GROUP 
1  2         3 


lOOMyUS 


^      k ^^-o.4;uS-^^      I         (D)  PULSE  GROUPS  (EXPANDED) 

■     '       |300M/US|  I       I 


I 


(E)  TIMING   WAVE  (40MC)  EXPANDED 


0 
TIME 


Fig.  3  —  Timing  events  in  the  circulating  loop. 


72  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 

groups  rather  than  individual  pulses  through  the  system.  These  were 
derived  from  the  pulse  group  generator  which  is  capable  of  delivering 
any  number  up  to  5  pulses  for  each  short  input  pulse.  These  pulses  are 
about  15  milli-microseconds  long  at  the  base  and  spaced  25  milli-micro- 
seconds  apart.  The  amplitude  of  each  of  these  pulses  can  be  adjusted 
independently  to  any  value  from  zero  to  full  amplitude  making  it  pos- 
sible to  set  up  any  combination  of  the  five  pulses.  These  are  the  pulses 
which  are  used  to  gate,  or  modulate,  the  output  of  the  4-kmc  oscillator. 

The  total  delay  around  the  waveguide  loop  including  TW  tubes,  etc.,' 
was  0.4)usec  or  400  milli-microseconds.  This  was  sufficient  to  allow  time 
between  pulse  groups  and  yet  short  enough  that  groups  could  circulate 
24  times  in  the  available  9.8jLtsec  interval.  This  can  be  seen  from  Figs. 
3C  and  3D.  The  latter  figure  shows  an  expanded  view  of  circulating 
pulse  groups.  The  pulses  in  Group  1  are  inserted  into  the  loop  at  the 
beginning  of  each  gating  cycle,  the  remaining  groups  result  from  circu- 
lation around  the  loop. 

When  all  five  pulses  are  present  in  the  pulse  groups  the  pulse  repeti- 
tion frequency  is  40  mc.  (Pulse  interval  25  milli-microseconds).  For  this 
condition  timing  pulses  should  be  supplied  to  the  regenerator  at  the  rate 
of  40  million  per  second.  These  pulses  are  supplied  continuously  and  not 
in  groups  as  is  the  case  with  the  circulating  pulses.  See  Fig.  BE.  In  order 
to  maintain  time  coincidence  between  the  circulating  pulses  and  the  tim- 
ing pulses  the  delay  around  the  loop  must  be  adjusted  to  be  an  exact 
multiple  of  the  pulse  spacing.  In  this  experiment  the  loop  delay  is  equal 
to  16-pulse  intervals.  Since  timing  pulses  are  obtained  by  harmonic 
generation  from  the  quenching  frequency  as  will  be  discussed  later  this 
frequency  must  be  an  exact  submultiple  of  pulse  repetition  frequency. 
In  this  experiment  the  ratio  is  512  to  1. 

Although  the  above  discussion  is  based  on  a  five-pulse  group  and 
40-mc  repetition  frequency  it  turned  out  that  for  most  of  the  experi- 
ments described  here  it  was  preferable  to  drop  out  every  other  pulse, 
leaving  three  to  a  group  and  resulting  in  a  20-mc  repetition  frequency. 
The  one  exception  to  this  is  the  limited-band-width  experiment  which 
will  be  described  later.  - 

For  all  of  the  experiments  described  here  timing  pulses  were  derived 
from  the  78-kc  quenching  frequency  by  harmonic  generation.  A  pulse 
with  a  width  of  25  milli-microseconds  and  with  a  78-kc  repetition  fre- 
quency as  shown  in  Fig.  3B  supplied  the  input  to  the  timing  wave  gen- 
erator. This  generator  consists  of  several  stages  of  limiting  amplifiers  all 
tuned  to  20  mc,  followed  by  a  locked-in  20-mc  oscillator.  The  output  of 
the  amplifier  consists  of  a  train  of  20-mc  sine  waves  with  constant  ampli- 


til 


REGENERATION   OF   BINARY   MICROWAVE   PULSES  73 

tude  for  most  of  the  12.8Msec  period  but  falling  off  somewhat  at  the  end 
of  the  period.  This-train  locks  in  the  oscillator  which  oscillates  at  a  con- 
stant amplitude  over  the  whole  period  and  at  a  frequency  of  20  mc. 
Timing  pulses  obtained  from  the  cathode  circuit  of  the  oscillator  tube 
pro^'ided  the  timing  waves  for  most  of  the  experiments.  For  the  experi- 
ment where  a  40-mc  timing  wave  was  required  it  was  obtained  from  the, 
20  mc  train  by  means  of  a  frequency  doubler.  For  this  case  it  is  necessary 
for  the  output  of  the  timing  wave  generator  to  remain  constant  in  ampli- 
tude and  fixed  in  phase  for  the  512-pulse  interval  between  synchronizing 
pulses. 

In  spite  of  the  stringent  requirements  placed  upon  the  timing  equip- 
ment it  functioned  well  and  maintained  synchronism  over  adequately 
long  periods  of  time  without  adjustment. 

PERFORMANCE    OF   REGENERATOR 

Performance  of  the  regenerator  under  various  conditions  is  recorded 
on  the  accompanying  illustrations  of  recovered  pulse  envelopes.  The 
first  experiment  was  to  determine  the  effects  of  disturbances  which  arise 
at  only  one  point  in  a  system.  Such  effects  were  simulated  by  adding 
disturbances  along  with  the  group  of  pulses  as  they  were  fed  into  the 
circulating  loop  from  the  modulator.  This  is  equivalent  to  having  them 
occur  at  only  the  first  repeater  of  the  chain. 

Some  of  the  first  experiments  also  involved  the  use  of  extraneous 
pulses  to  represent  noise  or  distortion  since  these  pulses  could  be  syn- 
chronized and  thus  studied  more  readily  than  could  random  effects.  In 
,  Fig.  4A  the  first  pulse  at  the  left  represents  a  desired  digit  pulse  with 
'  its  amplitude  increased  by  a  burst  of  noise,  the  second  pulse  represents 
'  a  clean  digit  pulse,  and  the  third  pulse  a  burst  of  noise.  This  group  is  at 
1  the  input  to  the  regenerator.  Fig.  4B  shows  the  same  group  of  pulses 
'  after  traversing  the  regenerator  once.  The  pulses  are  seen  to  be  shortened 
due  to  the  gating,  or  retiming,  action.  There  is  also  seen  to  be  some  ampli- 
tude correction,  i.e.  the  two  desired  pulses  are  of  more  nearly  the  same 
j  amplitude  and  the  undesired  pulse  has  been  reduced  in  relative  ampli- 
tude. After  a  few  trips  through  the  regenerator  the  pulse  group  was 
rendered  practically  perfect  and  remained  so  for  the  rest  of  the  twenty- 
four  trips  around  the  loop.  Fig.  4C  shows  the  group  after  24  trips.  In 
'another  experiment  pulses  were  circulated  for  100  trips  without  deteri- 
oration. Nothing  was  found  to  indicate  that  regeneration  could  not  be 
repeated  indefinitely. 
Figs.  5 A  and  5B  represent  the  same  conditions  as  those  of  4 A  and  4B 


74  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


Fig.  4  —  Effect  of  regeneration  on  disturbances  which  occur  at  only  one  re- 
peater. A  —  Input  to  regenerator,  original  signal.  B  —  Output  of  regenerator, 
first  trip.  C  —  Output  of  regenerator,  24th  trip. 


Fig.  5  —  l']ffect  of  regeneration  on  disturbances  which  occur  at  only  one  re- 
peater. A  —  Input  to  regenerator,  first  four  groups.  B  —  Output  of  regenerator, 
first  four  groups.  C  —  Output  of  regenerator,  increased  input  level. 


REGENERATION   OF   BINARY   MICROWAVE   PULSES 


75 


Fig.  6  —  Effect  of  regeneration  on  disturbances  which  occur  at  only  one  re- 
peater. A  —  Input  to  regenerator,  original  signal.  B — ^  Output  of  regenerator, 
first  trip.  C  • —  Oi^tput  of  regenerator,  24th  trip. 


except  that  the  oscilloscope  sweep  has  been  contracted  in  order  to  show 
the  progressive  effects  produced  by  repeated  passage  of  the  signal  through 
the  regenerator.  Fig.  5B  shows  that  after  the  pulses  have  passed  through 
the  regenerator  only  twice  all  visible  effects  of  the  disturbances  have 
been  removed.  Fig.  5C  shows  the  effect  of  simply  increasing  the  RF 
pulse  input  to  the  regenerator  by  approximately  4  db.  The  small  "noise" 
pulse  which  in  the  previous  case  was  quickly  dropped  out  because  of 
being  below  the  slicing  level  has  now  come  up  above  the  slicing  level 
and  so  builds  up  to  full  amplitude  after  only  a  few  trips  through  the 
regenerator.  Note  that  in  the  cases  shown  in  Figs.  4  and  5  discrimination 
against  unwanted  pulses  has  been  purely  on  an  amplitude  basis  since 
the  gate  has  been  unblocked  to  pulses  with  amplitudes  above  the  slicing 
level  whenever  one  of  these  distiu'bing  pulses  was  present. 

For  Fig.  6A  conditions  are  the  same  as  for  Fig.  4A  except  that  an  ad- 
ditional pulse  has  been  added  to  simulate  intersymbol  noise  or  inter- 
ference. Fig.  6B  indicates  that  after  only  one  trip  through  the  regenerator 
the  effect  of  the  added  pulse  is  very  small.  After  a  few  trips  the  effect 
is  completely  eliminated  leaving  a  practically  perfect  group  which  con- 
tinues on  for  24  trips  as  shown  by  Fig.  6C.  For  the  intersymbol  pulse, 
discrimination  is  on  a  time  basis  since  this  interference  occurs  at  a  time 


76 


THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


Fig.  7  —  Effect  of  regenerating  in  amplitude  without  retiming.  A  —  Outputof 
regenerator,  no  timing,  firt  trip.  B  —  Output  of  regenerator,  no  timing,  10th  trip. 
Output  of  regenerator,  no  timing,  23rd  trip. 

when  no  gating  pulse  is  present  and  hence  finds  the  gate  blocked  regard- 
less of  amplitude. 

To  show  the  need  for  retiming  the  pictures  shown  on  Figs.  7  and  8 
were  taken.  These  were  taken  with  the  amplitude  slicer  in  operation  but 
with  the  pulses  not  being  retimed.  Figs.  7A,  7B  and  7C,  respectively, 
show  the  output  of  the  slicer  for  the  first,  tenth  and  twenty-third  trips. 
After  ten  trips,  there  is  noticeable  time  jitter  caused  by  residual  noise 
in  the  system;  after  23  trips  this  jitter  has  become  severe  though  pulses 
are  still  recognizable.  It  should  be  pointed  out  that  for  this  experiment 
no  noise  was  purposely  added  to  the  system  and  hence  the  signal-to- 
noise  ratio  was  much  better  than  that  which  would  probably  be  encoun- 
tered in  an  operating  system.  For  such  a  system  we  would  expect  time 
jitter  effects  to  build  up  much  more  rapidly.  For  Fig.  8  conditions  are 
the  same  as  for  Fig.  7  except  that  the  pulse  spacing  is  decreased  by  the 
addition  of  an  extra  pulse  at  the  input.  Now,  after  ten  trips,  time  jitter 
is  bad  and  after  23  trips  the  pulse  group  has  become  little  more  than  a 
smear.  This  increased  distortion  is  probably  due  to  the  fact  that  less 
jitter  is  now  required  to  cause  overlap  of  pulses.  There  may  also  be  some 
effects  due  to  change  of  duty  cycle.  For  Fig.  9  there  was  neither  slicing 
nor  retiming  of  pulses.  Here,  pulse  groups  deteriorate  very  rapidly  to 
nothing  more  than  blobs  of  energy.  Note  that  there  is  an  increase  of 


i 


REGENERATION    OF   BINARY   MICROWAVE    PULSES 


77 


Fig.  8  — ■  Effect  of  regenerating  in  amplitude  without  retiming.  A  —  Output  of 
regenerator,  no  timing,  first  trip.  B  —  Output  of  regenerator,  no  timing,  10th 
trip.  C  —  Output  of  regenerator,  no  timing,  23rd  trip. 


Fig.  9  —  Pulses  circulating  through  the  loop  without  regeneration.  A  —  Origi- 
nal input.  B  —  4th  trip  without  regeneration.  C  —  20th  to  24th  trip  without  re- 
generation. 


'8 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


iWWWMMMWWIWWMilJflM^     II        .    rlilllT-     i  \m....:    iniTiinr-     IH. 


Fig.  10  —  The  regeneration  of  band-limited  pulses.  A  —  Input  to  regenerator, 
first  two  groups.  B  —  Output  of  regenerator,  first  two  groups.  C  —  Output  of 
regenerator,  24th  trip. 

amplitude  with  each  trip  around  the  loop  indicating  that  loop  gain  was 
slightly  greater  than  unity.  Without  the  sheer  it  is  difficult  to  set  the 
gain  to  exactly  unity  and  the  amplitude  tends  to  either  increase  or  de-  : 
crease  depending  upon  whether  the  gain  is  greater  or  less  than  unity. 
Results  indicated  by  the  pictures  of  Fig  9  are  possibly  not  typical  of  a 
properly  functioning  system  but  do  show  what  happened  in  this  par-  . 
ticular  sj^stem  when  regeneration  was  dispensed  with. 

Another  important  function  of  regeneration  is  that  of  overcoming  . 
band-limiting  effects.  Figs.  10  and  11  show  what  can  be  accomplished.  . 
For  this  experiment  the  pulse  groups  inserted  into  the  loop  were  as  shown  i| 
at  the  left  in  Fig.  lOA.  These  pulses  were  15  milli-microseconds  wide  at 
the  base  and  spaced  by  25  milli-microseconds  which  corresponds  to  a  j 
repetition  frequency  of  40  mc.  After  passing  through  a  band-pass  filter 
these  pulses  were  distorted  to  the  extent  shown  at  the  right  in  Fig.  lOA. 
From  the  characteristic  of  the  filter,  as  shown  on  Fig.  12,  it  is  seen  that 
the  bandwidth  employed  is  not  very  different  from  the  theoretical  min- 
imum required  for  double  sideband  transmission.  This  minimum  char- 
acteristic is  shown  by  the  dashed  lines  on  Fig.  12.  Fig.  lOB  shows  that 
at  the  output  of  the  regenerator  the  effects  of  band  limiting  have  been 
removed.  This  is  borne  out  by  Fig.  IOC  which  shows  that  after  24  trips 
the  code  group  was  still  practically  perfect.  It  should  l)e  pointed  out 
that  the  pulses  traversed  the  filter  once  for  each  trip  around  the  loop, 


REGENERATION    OF   BINARY   MICROWAVE   PULSES 


79 


Fig.  11  —  The  regeneration  of  band-limited  pulses.  A  —  Input  to  regenerator, 
first  two  groups.  B  —  Output  of  regenerator,  first  two  groups.  C  —  Output  of  re- 
generator, 24th  trip. 


that  is  for  each  trip  the  input  to  the  regenerator  was  as  shown  at  the  right 
of  Fig.  lOA  and  the  output  as  shown  by  Fig.  lOB.  It  is  important  to 
note  that  Fig.  12  represents  the  frequency  characteristic  of  a  single  hnk 
of  the  simulated  system.  The  pictures  of  Fig.  11  show  the  same  experi- 
ment but  this  time  with  a  different  code  group.  Any  code  group  which 
we  could  set  up  with  our  five  digit  pulses  was  transmitted  equally  well. 
In  order  to  determine  the  breaking  point  of  the  experimental  system, 
broad-band  noise  obtained  from  a  traveling-wave  amplifier  was  added 
into  the  system  as  shown  on  Fig.  2.  The  breaking  point  of  the  system  is 
the  noise  level  which  is  just  sufficient  to  start  producing  errors  at  the 
output  of  the  system.*  The  noise  is  seen  to  be  band-limited  in  exactly 
the  same  way  as  the  signal.  With  the  system  adjusted  to  operate  properly 
the  level  of  added  noise  was  increased  to  the  point  where  errors  became 
barely  discernible  after  24  trips  around  the  loop.  Noise  level  was  now 
reduced  slightly  (no  errors  discernible)  and  the  ratio  of  rms  signal  to  rms 
noise  measured.  Fig.  13A  shows  the  input  to  the  regenerator  for  the  23rd 
and  24th  trips  with  this  amount  of  noise  added.  Note  that  the  noise  has 

*  The  type  of  noise  employed  has  a  Gaussian  amplitude  distribution  and  there- 
fore there  was  actually  no  definite  breaking  point  —  the  rate  at  which  errors  Oc- 
curred increased  continuously  as  noise  amplitude  was  increased.  The  breaking 
point  was  taken  as  the  noise  level  at  which  errors  became  barely  discernible  on 
the  viewing  oscilloscope.  More  accurate  measurements  made  in  other  experiments 
indicate  that  this  is  a  fairly  satisfactory  criterion. 


80  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

28 


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FREQUENCY    IN    MEGACYCLES   PER   SECOND 

Fig.  12  —  Characteristics  of  the  band-pass  microwave  filter. 


m     % 


I 


JYYYYYYTin 


Fig.  13.  — The  regeneration  of  pulses  in  the  presence  of  broad-hand,  random 
noise  added  at  each  repeater.  A  —  Ini)ut  to  regenerator,  23rd  and  24th  trijis, 
broad-band  noise  added.  B  —  Ini)ut  to  regenerator,  23rd  and  24th  trips,  no  added 
noise.  C  —  20-mc  timing  wave. 


\ 


KEGENERATION  OF  BINARY  MICROWAVE  PULSES 


81 


Fig.  14  —  The  regeneration  of  pulses  in  the  presence  of  interference  occurring 
at  each  repeater.  A  —  Original  signal  with  added  moduhited  carrier  interference. 
B  —  Input  to  regenerator,  24th  trip,  niochilatod  carrier  interference.  C  —  Output 
of  regenerator,  24th  trip,  modulated  carrier  interference. 


produced  a  considerable  broadening  of  the  oscilloscope  trace.  Fig.  13B 
shows  the  same  pulse  groups  with  no  added  noise.  These  photographs  are 
included  to  give  some  idea  as  to  how  bad  the  noise  was  at  the  l;)reaking 
point  of  the  system.  Of  course  maximum  noise  peaks  occur  rather  infre- 
quently and  do  not  show  on  the  photograph.  At  the  output  of  the  re- 
generator effects  due  to  noise  were  barely  discernible.  This  output  looked 
so  much  like  that  shown  at  Fig.  14C  that  no  separate  photograph  is 
shown  for  it. 

Figs.  14A,  14B  and  14C  show  the  effects  of  a  different  type  of  inter- 
ference upon  the  system.  This  disturbance  was  produced  by  adding  into 
the  system  a  carrier  of  exactly  the  same  frequency  as  the  signal  carrier 
(4  kmc)  but  modulated  by  a  14-mc  wave,  a  frequency  in  the  same  order 
as  the  pulse  rate.  Here  again  the  level  of  the  interference  was  adjusted 
to  be  just  below  the  l)reaking  point  of  the  system.  A  comparison  between 
Figs.  14B  and  14C  gives  convincing  evidence  that  the  regenerator  has 
substantially  restored  the  waveform. 

For  the  case  of  the  interfering  signal  a  ratio  of  signal  to  interference 
of  10  db  on  a  peak-to-peak  basis  was  measured  when  the  interference 
was  just  below  the  breaking  point  of  the  system.  This,  of  course,  is  4  db 
above  the  theoretical  value  for  a  perfect  regenerator.  For  the  case  of 


82  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

broad-band  random  noise  an  rms  signal  to  noise  ratio  of  20  dl)  was  meas- 
ured.* This  compares  Avith  a  ratio  of  18  db  as  measured  by  Messrs. 
Meacham  and  Peterson  for  a  system  employing  complete  regeneration 
and  a  single  repeater,  f 

Recently,  A.  F.  Dietrich  repeated  the  circulating  loop  experiment  at 
a  radio  frequency  of  11  kmc.  His  determinations  of  required  signal-to- 
noise  ratios  are  substantially  the  same  as  those  reported  here.  From  the 
various  experiments  we  conclude  that  for  a  long  chain  of  properly  func- 
tioning regenerative  repeaters  of  i-he  type  discussed  here  practically 
perfect  transmission  is  obtained  as  long  as  the  signal-to-noise  ratio  at 
the  input  to  each  repeater  is  20  db  or  better  on  an  rms  basis.  In  an  operat- 
ing system  it  might  be  desirable  to  increase  this  ratio  to  23  db  to  take 
care  of  deficiencies  in  automatic  gain  controls,  power  changes,  etc. 

From  the  experiments  we  also  conclude  that  the  price  we  pay  for  using 
partial  instead  of  complete  regeneration  is  about  3  to  4  db  increase  in 
the  required  signal-to-noise  ratio.  In  a  radio  system  which  provides  a 
fading  margin  this  penalty  would  be  less  since  the  probability  that  two 
or  more  adjacent  links  will  reach  maximum  fades  simultaneously  is  very  ' 
small.  Under  these  conditions  only  one  repeater  at  a  time  would  be  near 
the  breaking  point  and  the  system  would  behave  much  as  though  the 
repeater  provided  complete  regeneration. 

TIMING 

Although  we  have  considered  the  problem  of  retiming  of  signal  pulses 
up  to  now  we  have  not  discussed  the  problem  of  obtaining  the  necessary  ' 
timing  pulses  to  perform  this  function,  but  have  simpl}^  assumed  that  a 
source  of  such  pulses  was  available.  As  w^as  mentioned  earlier  timing   I 
pulses  would  probably  be  derived  from  the  signal  pulses  in  a  practical  »^ 
system.  These  pulses  would  be  fed  into  some  narrow  band  amplifier 
tuned  to  pulse  repetition  frequency.  The  output  of  this  circuit  could  be 
made  to  be  a  sine  wave  at  repetition  frequency  if  gaps  between  the  input 
pulses  were  not  too  great.  Timing  pulses  could  be  derived  from  this  sine 
wave.  This  timing  equipment  could  be  similar  to  that  used  in  these  ex- 
periments and  described  earlier.  Further  study  of  the  problems  of  ob- 
taining timing  information  is  being  made. 

*  For  Gaussian  noise  it  is  not  possible  to  specif.y  a  theoretical  value  of  minimum 
S/N  ratio  without  specifying  the  tolerable  percentage  of  errors.  For  the  number  of 
errors  detectable  on  the  oscilloscope  it  seems  rasonable  to  assume  a  12  db  peak 
factor  for  the  noise.  The  peak  factor  for  the  signal  is  3  db.  The  6  db  peak  S/N 
which  would  be  required  for  an  ideal  regenerator  then  becomes  15  db  on  an  rms 
basis. 

t  L.  A.  Meacham  and  E.  Peterson,  B.  S.  T.  J.,  p.  43,  Jan.,  1948. 


" 


KEGENERATION   OF   BINARY   MICROWAVE   PULSES 


83 


'  GATING 
PULSE 


INPUT 


OUTPUT 


Fig.  15A  —  Low-frequency  equivalent  of  the  partial  regenerator. 


DESCRIPTION    OF   REGENERATOR 

This  device  regenerates  pulses  by  performing  on  them  the  operations 
of  ''slicing"  and  retiming. 
An  ideal  slicer  is  a  device  with  an  input-output  characteristics  such  as 
shown  by  the  dashed  lines  of  Fig.  15C.  It  is  seen  that  for  all  input  levels 
below  the  so-called  slicing  level  transmission  through  the  device  is  zero 
but  that  for  all  amplitudes  greater  than  this  value  the  output  level  is 
finite  and  constant.  Thus,  all  input  voltages  which  are  less  than  the  slic- 
ing level  have  no  effect  upon  the  output  whereas  all  input  voltages 
greater  than  the  slicing  level  produce  the  same  amplitude  of  output. 
Normally  conditions  are  adjusted  so  that  the  slicing  level  is  at  one-half 


INPUT    LEVEL 


Fig.  15B  —  Characteristics  of  the  separate  branches  with  ditterential  bias. 


84 


THE    BELL    SYSTEM   TECHNICAL  JOURNAL,    JANUARY    1956 


INPUT    LEVEL 


Fig.  15C  —  Resultant  output  with  differential  bias. 


BRANCH  2 
BRANCH  1 


RESULTANT 


INPUT   LEVEL 


Fig.  15D  —  Characteristics  of  the  separate  branches  and  resultant  output  with 
equal  biases. 


of  peak  pulse  amplitude  —  then  at  the  output  of  the  slicer  there  will  be 
no  effect  whatsoever  from  disturbances  unless  these  disturbances  exceed 
half  of  the  pulse  amplitude.  It  is  this  slicing  action  which  removes  the 
amplitude  effects  of  noise.  Time  jitter  effects  are  removed  by  retiming, 
i.e.,  the  device  is  made  to  have  high  loss  regardless  of  input  level  except 
at  those  times  when  a  gating  pulse  is  present. 

Fig.  15A  shows  schematically  a  low-frequency  equivalent  of  the  re- 
generator used  in  these  experiments.  Here  an  input  line  divides  into  two 
identical  branches  isolated  from  each  other  and  each  with  a  diode  shunted 
across  it.  The  outputs  of  the  two  branches  are  recombined  through  neces- 
sary isolators  to  form  a  single  output.  The  phase  of  one  branch  is  re- 
versed before  recombination,  so  that  the  final  output  is  the  difference 
between  the  two  individual  outputs. 

Fig.  15B  shows  the  input-output  characteristics  of  the  two  branches 
when  the  diodes  are  biased  back  to  be  non-conducting  by  means  of  bias 
voltages  Vi  and  V2  respectively.  For  low  levels  the  input-output  char- 
acteristic of  both  branches  will  be  linear  and  have  a  45°  slope.  As  soon 


REGENEKATION   OF   BINARY    MICROWAVE   PULSES 


85 


as  the  input  voltage  in  a  branch  reaches  a  vakie  equal  to  that  of  the  back 
bias  the  diode  will  start  to  conduct,  thus  absorbing  power  and  decrease 
the  slope  of  the  characteristic.  The  output  of  Branch  1  starts  to  flatten 
off  when  the  input  reaches  the  value  Vi  ,  while  the  output  of  Branch  2 
does  not  flatten  until  the  input  reaches  the  value  V2  .  The  combined 
output,  which  is  equal  to  the  differences  of  the  two  branch  outputs,  is 
then  that  shown  by  the  solid  line  of  Fig.  15C  and  is  seen  to  have  a  transi- 
tion region  between  a  low  output  and  a  high  output  level.  If  the  two 
branches  are  accurately  balanced  and  if  the  signal  voltage  is  large  com- 
pared to  the  differential  bias  V2  —  Vi  the  transition  becomes  sharp  and 
the  device  is  a  good  slicer. 

If  the  two  diodes  are  equally  biased  as  shown  on  Fig.  15D  the  outputs 
of  the  two  branches  should  be  nearly  equal  regardless  of  input  and  the 
total  output,  which  is  the  difference  between  the  two  branch  outputs, 
will  always  be  small. 

Fig.  16  shows  a  microwave  equivalent  of  the  circuit  of  Fig.  15A.  In 
the  microwave  structure  lengths  of  wave-guide  replace  the  wire  lines  and 
branching,  recombining  and  isolation  are  accomplished  by  means  of 
hybrid  junctions.  The  hybrid  shown  here  is  of  the  type  known  as  the  lA 
junction. 

Fig.  17  shows  another  equivalent  microwave  structure  employing  only 
one  hybrid.  This  is  the  type  used  in  the  experiments  described  here.  The 
[output  consists  of  the  combined  energies  reflected  from  the  two  side 
jarms  of  the  junction.  With  the  junction  connected  as  shown  phase  rela- 
Itionships  are  such  that  the  output  is  the  difference  between  the  reflec- 


GATING 
PULSE 


^(— r-V\^^^ 


RF 
INPUT  ARM 


PROBE 


TERMINATION 

I 


ARM  4 


I— vw-^ 


Fig.  16  —  Microwave  regenerator. 


86 


THE    BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 


tions  from  the  two  side  arms  so  that  when  conditions  in  the  two  arms 
are  identical  there  is  no  output.  The  crystal  diodes  coupled  to  the  side 
arms  are  equivalent  to  those  shunted  across  the  two  lines  of  Fig.  15A. 

Fig.  18,  which  is  a  plot  of  the  measured  input-output  characteristic 
of  the  regenerator  used  in  the  loop  test,  shows  how  the  device  acts  as  a 
combined  sheer  and  retimer.  Curve  A,  ol)tained  with  equal  biases  on  the 
two  diodes,  is  the  characteristic  with  no  gating  pulse  applied  i.e.  the 
diodes  are  normally  biased  in  this  manner.  It  is  seen  that  this  condition 
produces  the  maximum  of  loss  through  the  device.  By  shifting  one  diode 
bias  so  as  to  produce  a  differential  of  0.5  volt  the  characteristic  changes 
to  that  of  Curve  B.  This  differential  bias  can  be  supplied  by  the  timing 
pulse  in  such  a  way  that  this  pulse  shifts  the  characteristic  from  that 
shown  at  A  to  that  shown  at  B  thus  decreasing  the  loss  through  the  de- 
vice by  some  12  to  15  db  during  the  time  the  pulse  is  present.  In  this  way 
the  regenerator  is  made  to  act  as  a  gate  —  though  not  an  ideal  one. 

We  see  from  curve  B  that  with  the  differential  bias  the  device  has  the 
characteristic  of  a  slicer  —  though  again  not  ideal.  For  lower  levels  of 
input  there  is  a  region  over  which  the  input-output  characteristic  is 
square  law  with  a  one  db  change  of  input  producing  a  two  db  change  of 
output.  This  region  is  followed  by  another  in  which  limiting  is  fairly 
pronounced.  At  the  8-db  input  level,  which  is  the  point  at  which  limiting 
sets  in,  the  loss  through  the  regenerator  was  measured  to  be  approxi- 
mately 12  db.  The  characteristic  shown  was  found  to  be  reproducible 
both  in  these  experiments  at  4  kmc  and  in  those  bj'-  A.  F.  Dietrich  at 
11  kmc. 

For  a  perfect  slicer  only  an  infinitesimal  change  of  input  level  is  re- 


GATING 
PULSE 


■AAV-i_ 


ARM   2 


RF 

OUTPUT 


Fig,  17  —  Microwave  regenerator  employing  a  single  hybrid  junction. 


REGENERATION    OF   BINARY   MICROWAVE   PULSES 


87 


ID 

m 
o 

LU 

a 


D 


3 

o 


-10 


-12 


-14 


-16 


-18 


-20 


-22 


-24 


V,  =  0.5 
V2  =  0 

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12  DB 

LOSS 

i>— <! 

P'< 

JH   " 

^ 

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^ 

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1 

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( 
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1 
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6  8  10  12 

INPUT  LEVEL    IN    DECIBELS 


14 


16 


18 


Fig.  18  —  Static  characteristics  of  the  regenerator  employed  in  these  experiments. 

f}uired  to  change  the  output  from  zero  to  maximum.  The  input  level  at 
which  this  transition  takes  place  is  the  slicing  level  and  has  a  very  defi- 
nite value.  For  a  characteristic  such  as  that  shown  on  Fig.  18  this  point 
is  not  at  all  definite  and  the  question  arises  as  to  how  one  determines  the 
slicing  level  for  such  a  device.  Obviously  this  point  should  be  somewhere 
on  the  portion  of  the  characteristic  where  expansion  takes  place.  In  the 
case  of  the  circulating  loop  the  slicing  level  is  the  level  for  which  total 
gain  around  the  loop  is  exactly  etiual  to  unity.  Why  this  is  so  can  be  seen 
from  Fig.  19  which  is  a  plot  of  gain  \'ersus  input  level  for  a  repeater 
containing  a  sheer  with  a  characteristic  as  shown  by  curve  B  of  Fig.  18. 
Amplifiers  are  necessary  in  the  loop  to  make  up  for  loss  through  the  re- 
generator and  other  components.  For  Fig.  11)  we  assume  that  these 
amplifiers  have  been  adjusted  so  that  gain  around  the  loop  is  exactly 
unity  for  an  input  pulse  having  a  peak  amplitude  corresponding  to  the 


88 


THE    BELL   SYSTEM    TECHNICAL   JOURNAL,    JANUARY    1956 


-3 


-2-1  0  1  2  3  4  5 

INPUT    LEVEL    IN    DECIBELS    ABOVE    SLICING    LEVEL 


Fig  19  —  Gain  characteristics  of  u  repeater  providing  partial  regeneration. 


point  F'  of  Fig.  18.  On  Fig.  19  all  other  levels  are  shown  in  reference  to 
this  unity-gain  value. 

From  Fig.  19  it  is  obvious  that  a  pulse  which  starts  out  in  the  loop 
with  a  peak  amplitude  exactly  equal  to  the  reference,  or  slicing  level, 
will  continue  to  circulate  without  change  of  amplitude  since  for  this 
level  there  is  unity  gain  around  the  loop.  A  pulse  with  amplitude  greater 
than  the  slicing  level  will  have  its  amplitude  increased  by  each  passage 
through  a  regenerator  until  it  eventually  reaches  a  value  of  +6  db.  It 
will  continue  to  circulate  at  this  amplitude,  for  here  also  the  gain  around 
the  loop  isVmity.*  Any  pulse  with  peak  amplitude  less  than  the  reference 
level  will  have  its  amplitude  decreased  by  successive  trips  through  the 
regenerator  and  eventually  go  to  zero.  We  also  see  that  the  greater  the 
departure  of  the  amplitude  of  a  pulse  from  the  slicing  level  the  more 
effect  the  regenerator  has  upon  it.  This  means  that  the  device  acts  much 
more  powerfully  on  low  level  noise  than  on  noise  with  pulse  peaks  near 
the  slicing  level.  As  examples  consider  first  the  case  of  noise  peaks  only 
1  db  below  slicing  level  at  the  input  (peak  S/N  =  7  db).  At  this  level 
there  is  a  1  db  loss  through  the  repeater  so  that  at  the  output  the  noise 
peaks  will  be  2  db  below  reference  to  give  a  *S/A^  ratio  of  8  db.  Next 


*  Note  that  llic  ^-fi-dl)  level  is  at  a  point  of  stable  equilibrium  whereas  at  the 
slicing  level  C(iuilil)rium  is  unstable. 


REGENERATION    OF   BINARY   MICROWAVE   PULSES  89 

consider  noise  with  a  peak  level  5  db  below  slicing  level  (S/N  =11  db) 
at  the  input.  The  loss  at  this  level  is  5  db  resulting  in  a  noise  level  10  db 
below  reference  to  give  a  S/N  ratio  of  16  db.  We  see  that  a  4  db  improve- 
ment in  S/N  ratio  at  the  input  results  in  an  8  db  improvement  in  this 
ratio  at  the  output. 

Everything  which  was  said  above  concerning  the  circulating  loop  ap- 
plies equally  to  a  chain  of  identical  repeaters.  To  set  the  effective  slicing 
level  at  half  amplitude  at  each  repeater  in  a  chain  one  would  first  find 
two  points  on  the  sheer  characteristics  such  as  P  and  P'  of  Fig.  18.  The 
point  P  should  be  in  the  region  of  expansion  and  P'  in  the  limiting  region. 
Also  the  points  should  be  so  chosen  that  a  6  db  increase  of  input  from 
that  at  point  P  results  in  a  6  db  increase  in  output  at  the  point  P'.  If 
now  at  each  repeater  we  adjust  pulse  peak  amplitude  at  the  sheer  input 
to  a  value  corresponding  to  that  at  point  P'  we  will  have  unity  gain 
from  one  repeater  to  the  next  at  levels  corresponding  to  pulse  peaks. 
We  will  also  have  unity  gain  at  levels  corresponding  to  one  half  of  pulse 
amplitude.  The  effective  slicing  level  is  thus  set  at  half  amplitude.  Ob- 
viously the  procedure  for  setting  the  slicing  level  at  some  value  other 
than  half  amplitude  would  be  practically  the  same.  It  should  be  pointed 
out  that  although  half  amplitude  is  the  preferred  slicing  level  for  base- 
band pulses  this  is  not  the  case  for  carrier  pulses.  W.  R.  Bennett  of  Bell 
Telephone  Laboratories  has  shown  that  for  carrier  pulses  the  probability 
that  noise  of  a  given  power  will  reduce  signal  pulses  below  half  amplitude 
is  less  than  the  probability  that  this  same  noise  will  exceed  half  ampli- 
tude. This  comes  about  from  the  fact  that  for  effective  cancellation  there 
must  be  a  180°  phase  relationship  between  noise  and  pulse  carrier.  For 
this  reason  the  slicing  level  should  be  set  slightly  above  half  amplitude 
for  a  carrier  pulse  system. 

The  difference  in  performance  between  a  perfect  sheer  and  one  with 
characteristics  such  as  shown  on  Fig.  18  are  as  follows:  For  the  perfect 
sheer  no  effects  from  noise  or  other  disturbances  are  passed  from  one 
repeater  to  the  next.  For  the  case  of  the  imperfect  regenerator  some  ef- 
fects are  passed  on  and  so  tend  to  accumulate  in  a  chain  of  repeaters. 
To  prevent  this  accumulated  noise  from  building  up  to  the  breaking 
point  of  the  system  it  is  necessary  to  make  the  signal-to-noise  ratio  at 
each  repeater  somewhat  better  than  that  which  would  be  required  with 
the  ideal  sheer.  For  the  case  of  random  noise  the  required  S/N  ratio 
seems  to  be  about  5  or  6  db  above  the  theoretical  value.  This  is  due  in 
part  to  sheer  deficiency  and  in  part  to  other  system  imperfections. 


90  THE    BELL    SYSTEM   TECHNICAL  JOURNAL,    JANUARY    19o() 

CONCLUSIONS 

It  is  possible  to  build  a  simple  device  for  regenerating  pulses  directly 
at  microwave  frequencies.  A  long  chain  of  repeaters  employing  this 
regenerator  should  perform  satisfactorily  as  long  as  the  rms  signal-to- 
noise  ratio  at  each  repeater  is  maintained  at  a  value  of  20  db  or  greater. 
There  are  a  number  of  remaining  problems  which  must  be  solved  before 
we  have  a  complete  regenerative  repeater.  Some  of  these  problems  are: 

(1)  Recovery  of  information  for  retiming  from  the  incoming  pulse  train; 

(2)  Automatic  gain  or  level  control  to  set  the  slicing  level  at  each  re- 
peater; (3)  Simple,  reliable,  economical,  broad-band  microwave  ampli- 
fiers. (4)  Proper  filters  —  both  for  transmitting  and  receiving.  Traveling- 
wave  tube  development  should  eventually  result  in  amplifiers  which 
will  meet  all  of  the  requirements  set  forth  in  (3)  above.  Any  improve- 
ments which  can  be  made  in  the  regenerator  without  adding  undue 
complications  would  also  be  advantageous. 

ACKNOWLEDGMENTS 

A.  F.  Dietrich  assisted  in  setting  up  the  equipment  described  here  and 
in  many  other  ways.  The  experiment  would  not  have  been  possible  with- 
out traveling-wave  tubes  and  amplifiers  which  were  obtained  through 
the  cooperation  of  M.  E.  Hines,  C.  C.  Cutler  and  their  associates.  I  wish 
to  thank  W.  M.  Goodall,  and  J.  R.  Pierce  for  many  valuable  suggestions. 


Crossbar  Tandem  as  a  Long  Distance 
Switching  System 

By  A.  O.  ADAM 

(Manuscript  received  March  4,  1955) 

Major  toll  switching  features  are  being  added  to  the  crossbar  tandem 
switching  system  for  use  at  many  of  the  important  long  distance  switching 
centers  of  the  nationwide  network.  These  include  automatic  selection  of  one 
of  several  alternate  routes  to  a  'particular  destination,  storing  and  sending 
forward  digits  as  required,  highly  flexible  code  conversion  for  transmitting 
digits  different  from  those  received,  and  a  translating  arrangement  to  select 
the  most  direct  route  to  a  destination.  The  system  is  designed  to  serve  both 
operator  and  customer  dialed  long  distance  traffic. 

INTRODUCTION 

The  crossbar  tandem  switching  system,^  originally  designed  for  switch- 
ing between  local  dial  offices,  will  now  play  an  important  role  in  nation- 
wide dialing.  New  features  are  now  available  or  are  being  developed  that 
will  permit  this  system  to  switch  all  types  of  traffic.  As  a  result,  crossbar 
[  tandem  offices  will  have  widespread  use  at  many  of  the  important  switch- 
ing centers  of  the  nationwide  switching  network. 

This  paper  briefly  reviews  the  crossbar  tandem  switching  system  and 
its  application  for  local  switching,  followed  by  discussion  of  the  general 
aspects  of  the  nationwide  switching  plan  and  of  the  major  new  features 
required  to  adapt  crossbar  tandem  to  this  plan. 

CROSSBAR  TANDEM  OFFICES  USED  FOR  LOCAL  SWITCHING 

Crossbar  tandem  offices  are  now  used  in  many  of  the  large  metropolitan 
areas  throughout  the  country  for  interconnecting  all  types  of  local  dial 
offices.  In  these  applications  they  perform  three  major  functions.  Basi- 
cally, they  permit  economies  in  trunking  by  combining  small  amounts  of 

91 


02  THE    BELL   SYSTEM   TECHXIf  AL   JOURNAL,   JANUARY    1956 

traffic  to  and  from  the  local  offices  into  larger  amounts  for  routing  over 
common  triuik  groups  to  gain  increased  efficiency  resulting  in  fewer  over- 
all trunks. 

A  second  important  function  is  to  permit  handling  calls  economically 
between  different  types  of  local  offices  which  are  not  compatible  from  the 
standpoint  of  intercommunication  by  direct  pulsing.  Crossbar  tandem 
offices  serve  to  connect  these  offices  and  to  supply  the  conversion  from 
one  type  of  pulsing  to  another  where  such  incompatibilities  exist. 

The  third  major  function  is  that  of  centralization  of  equipment  or 
services.  For  example,  centralization  of  expensive  charging  equipment  at 
a  crossbar  tandem  office  results  in  efficient  use  of  such  equipment  and 
over-all  lower  cost  as  compared  with  furnishing  this  equipment  at  each 
local  office  requiring  it.  Examples  of  such  equipment  are  remote  control 
of  zone  registration  and  centralized  automatic  message  accounting.^  Cen- 
tralization of  other  services  such  as  weather  bureau,  time-of-day  and 
similar  services  can  be  furnished. 

The  first  crossbar  tandem  offices  were  installed  in  1941  in  New  York, 
Detroit  and  San  Francisco.  These  offices  were  equipped  to  interconnect 
local  panel  and  No.  1  crossbar  central  offices  in  the  metropolitan  areas, 
and  to  complete  calls  to  manual  central  offices  in  the  same  areas.  The  war 
years  slowed  both  development  and  production  and  it  was  not  until  the 
late  40's  that  many  features  now  in  use  were  placed  in  service.  These 
later  features  enable  customers  in  step-by-step  local  central  offices  on  the 
fringes  of  the  metropolitan  areas  to  interconnect  on  a  direct  dialing  basis 
with  metropolitan  area  customers  in  panel,  crossbar,  manual  and  step- 
by-step  central  offices.  This  same  development  also  permitted  central 
offices  in  strictly  step-by-step  areas  to  be  interconnected  by  a  crossbar 
tandem  office  where  direct  interconnecting  was  not  economical.  Facilities 
were  also  made  available  in  the  crossbar  tandem  system  for  completing 
calls  from  switchboards  where  operators  use  dials  or  multifrequency  key 
pulsing  sets. 

Since  a  crossbar  tandem  office  usually  has  access  to  all  of  the  local 
offices  in  the  area  in  which  it  is  installed,  it  is  attractive  for  handling 
short  and  long  haul  terminating  traffic.  The  addition  of  toll  terminal 
equipment  at  Gotham  Tandem  in  New  York  City  in  1947  permitted 
operators  in  New  York  State  and  northern  New  Jersey  as  well  as  distant 
operators  to  dial  or  key  pulse  directly  into  the  tandem  equipment  for 
completion  of  calls  to  approximately  350  central  offices  in  the  New  York 
metropolitan  area.  This  method  of  completing  these  calls  without  the 
aid  of  the  inward  operators  was  a  major  advance  in  using  tandem  switch- 
ing ecjuipment  for  speeding  completion  of  out-of-town  calls. 


CROSSBAR   TANDEM   AS   A   TOLL   SWITCHING   SYSTEM 


93 


CROSSBAR  TANDEM  SWITCHING  ARRANGEMENT 

The  connections  in  a  crossbar  tandem  office  are  established  through 
crossbar  switches  mounted  on  incoming  trunk  link  and  outgoing  office 
link  frames  shown  on  Fig.  1.  The  connections  set  up  through  these 
switches  are  controlled  by  equipment  common  to  the  crossbar  tandem 
office  which  is  held  only  long  enough  to  set  up  each  individual  connec- 
tion. Senders  and  markers  are  the  major  common  control  circuits. 

The  sender's  function  is  to  register  the  digits  of  the  called  number, 
transmit  the  called  office  code  to  the  marker  and  then,  as  subsequently 
directed  by  the  marker,  control  the  outpulsing  to  the  next  office. 

The  marker's  function  is  to  receive  the  code  digits  from  the  sender 
for  translation,  return  information  to  the  sender  concerning  the  de- 
tails of  the  call,  select  an  idle  outgoing  trunk  to  the  called  destination 
and  close  the  transmission  path  through  the  crossbar  switches  from  the 
incoming  to  the  outgoing  trunk. 

GENERAL  ASPECTS  OF  NATIONWIDE  DIALING 

Operator  distance  dialing,  now  used  extensively  throughout  the 
country,  as  well  as  customer  direct  distance  dialing  are  based  on  the 
division  of  the  United  States  and  Canada  into  numbering  plan  areas, 
interconnected  by  a  national  network  through  some  225  Control  Switch- 
ing Points  (CSP's)  equipped  with  automatic  toll  switching  systems. 
^  An  essential  element  of  the  nationwide  dialing  program  is  a  universal 
numbering  plan^  wherein  each  customer  will  have  a  distinctive  number 
which  does  not  conflict  with  the  number  of  any  other  customer.  The 
method  employed  is  to  divide  the  United  States  and  Canada  geographi- 


INCOMING 

TRUNK    FROM 

ORIGINATING 

OFFICE 


TANDEM 

TRUNK 


TRUNK    LINK    FRAME 


9     ? 


TRUNK    LINK 
CONNECTOR 


SENDER    LINK 


SENDER    LINK 
CONTROL   CIRCUIT 


SENDER 


OFFICE    LINK    FRAME 


<?    9 


OFFICE    LINK 
CONNECTOR 


J  4_ 

MARKER 


CONNECTOR 


OUTGOING 
TRUNK 


MARKER 


Fig.  1  —  Crossbar  tandem  switching  arrangement. 


94  THE   BELL   SYSTEM  TECHNICAL  JOURNAL,   JANUARY    1956 

cally  into  more  than  100  numbering  plan  areas  and  to  give  each  of  these 
a  distinctive  three  digit  code  with  either  a  1  or  0  as  the  middle  digit. 
Each  numbering  plan  area  will  contain  500  or  fewer  local  central  offices 
each  of  which  will  be  assigned  a  distinctive  three-digit  office  code. 
Thus  each  of  the  telephones  in  the  United  States  and  Canada  will  have, 
for  distance  dialing  purposes,  a  distinct  identity  consisting  of  a  three 
digit  area  code,  an  office  code  of  two  letters  and  a  numeral,  and  a  sta- 
tion number  of  four  digits.  Under  this  plan,  a  customer  will  dial  7  digits 
to  reach  another  customer  in  the  same  numbering  area  and  10  digits  to 
reach  a  customer  in  a  different  numbering  area. 

A  further  reciuirement  for  nationwide  dialing  of  long  distance  calls  is 
a  fundamental  plan"*  for  automatic  toll  switching.  The  plan  provides  a 
systematic  method  of  interconnecting  all  the  local  central  offices  and 
toll  switching  centers  in  the  United  States  and  Canada.  As  shown  on 
Fig.  2,  several  local  central  offices  or  "end  offices"  are  served  by  a  single 
toll  center  or  toll  point  that  has  trunks  to  a  "home"  primary  center 
which  serves  a  group  of  toll  centers.  Each  primary  center,  has  trunks  to 
a  "home"  sectional  center  which  serves  a  larger  area  of  the  country. 
Similuj-ly,  the  entire  toll  dialing  territory  is  divided  into  eleven  very 
large  areas  called  regions,  each  having  a  regional  center  to  serve  all  the 
sectional  centers  in  the  region.  One  of  the  regional  centers,  probably 
St.  Louis,  Missouri,  will  be  designated  the  national  center.  The  homing 
arrangements  are  such  that  it  is  not  necessary  for  end  offices,  toll  centers, 
toll  points  and  primary  centers  to  home  on  the  next  higher  ranking 
office  since  the  complete  final  route  chain  is  not  necessary.  For  example, 
end  offices  may  be  served  directly  from  any  of  the  higher  ranking  switch- 
ing centers  also  shown  in  Fig.  2. 

Collectively,  the  national  center,  the  regional  centers,  the  sectional 
centers  and  the  primary  centers  will  constitute  the  control  switching 
points  for  nationwide  dialing.  The  basic  switching  centers  and  homing 
arrangements  are  illustrated  in  Fig.  3. 

TANDEM  CROSSBAR  FEATURES  FOR  NATIONWIDE  DIALING 

The  broad  objective  in  developing  new  features  for  crossbar  tandem 
is  to  provide  a  toll  switching  system  that  can  be  used  in  cities  where 
the  large  capacity  and  the  full  versatilit}^  of  the  No.  4  toll  crossbar 
switching  system-''  may  not  be  economical. 

The  application  of  crossbar  tandem  two-wire  switching  systems  at 
primary  and  sectional  centers  has  been  made  possible  by  the  extended 
use  of  high  speed  carrier  systems.  The  echoes  at  the  2-wire  crossbar 
tandem  switching  offices  can  be  effectively  reduced  by  providing  a  high 


CROSSBAR   TANDEM   AS   A   TOLL   SWITCHING   SYSTEM 


95 


office  balance  and  by  the  use  of  impedance  compensators  and  fixed  pads. 
A  well  balanced  two-wire  switching  system,  proper  assignment  of  inter- 
toll  trunk  losses,  and  the  use  of  carrier  circuits  with  high  speed  of  propa- 
gation will  permit  through  switching  Mdth  little  or  no  impairment  from 
an  echo  standpoint. 

The  new  features  for  crossbar  tandem  will  provide  arrangements 
necessary  for  operation  at  control  switching  points  (CSP's).  These  in- 
clude automatic  alternate  routing,  the  ability  to  store  and  send  forward 


TP 


e 


I      I        NC  =   NATIONAL    CENTER 
RC  =   REGIONAL    CENTER 
/\       SC  =   SECTIONAL    CENTER 
(      J      PC  =    PRIMARY    CENTER 

Fig.  2  —  Homing  arrangement  for  local  central  offices  and  toll  centers. 


TC  =   TOLL   CENTER 
TP   =   TOLL   POINT 
EG   =   END   OFFICE 


96 


CROSSBAR   TANDEM   AS   A   TOLL   SWITCHING    SYSTEM 


97 


digits  as  required,  highly  flexible  code  conversion  (transmitting  forward 

i  different  digits  for  the  area  or  office  code  instead  of  the  dialed  digits), 

prefixing  digits  ahead  of  the  called  office  code,  and  six-digit  translation. 

ALTERNATE  ROUTING 

The  control  switching  points  will  be  interconnected  by  a  final  or 
"backbone"  network  of  intertoll  trunks  engineered  so  that  very  few 
calls  will  be  delayed.  In  addition,  direct  circuits  between  individual 
switching  offices  of  all  classes  will  be  provided  as  warranted  by  the 
traffic  density.  These  are  called  "high-usage"  groups  and  are  not  en- 
gineered to  handle  all  the  traffic  offered  to  them  during  the  busy  hour. 
Traffic  offered  to  a  high-usage  group  which  finds  all  trunks  busy  will  be 
automatically  rerouted  to  alternate  routes®-^  consisting  of  other  high- 
usage  groups  or  to  the  final  trunk  group.  The  abi.ity  of  the  crossbar 
tandem  equipment  at  the  control  switching  point  to  select  one  of  several 
alternate  routes  automatically,  when  all  choices  in  the  first  route  are 
busy,  contributes  to  the  economy  of  the  plant  and  provides  additional 
protection  against  complete  interruption  of  service  when  all  circuits  on 
a  particular  route  are  out  of  service. 

Fig.  4  shows  a  hypothetical  example  of  alternate  routing  when  a 
crossbar  tandem  office  at  South  Bend,  Indiana,  receives  a  call  destined 
for  ^Youngstown,  Ohio.  To  select  an  idle  path,  using  this  plan,  the 
switching  equipment  at  South  Bend  first  tests  the  direct  trunks  to 
Youngstown.  If  these  are  all  busy,  it  tests  the  direct  trunks  to  Cleveland 
where  the  call  would  be  completed  over  the  final  group  to  Youngstown. 
If  the  group  to  Cleveland  is  also  busy,  South  Bend  would  test  the  group 


CHICAGO 


SOUTH    BEND 

CROSSBAR 

TANDEM 


CLEVELAND 


-YOUNGSTOWN 


ITT5BURGH 


Fig.  4  —  Toll  network  —  alternate  routing. 


98 


THE    BELL   SYSTEM   TECHNICAL  JOURNAL,   JANUARY    1956 


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CROSSBAR   TANDEM   AS   A   TOLL   SWITCHING   SYSTEM  99 

to  Pittsburgh  and  on  its  last  attempt  it  would  test  the  final  group  to 
Indianapolis.  If  the  call  were  routed  to  Pittsburgh  or  Indianapolis,  the 
switching  equipment  at  these  points  would  attempt  by  first  choice  and 
alternate  routes  to  reach  Youngstown.  The  final  choice  backbone  route 
would  be  via  Indianapolis,  Chicago,  St.  Louis,  Pittsburgh,  Cleveland  to 
Youngstown.  Should  all  the  trunks  in  any  of  the  final  groups  tested  be 
busy  no  further  attempt  to  complete  the  call  is  made.  It  is  unlikely 
that  so  many  alternate  routes  would  be  provided  in  actual  practice 
since  crossbar  tandem  can  test  only  a  maximum  of  240  trunks  on  each 
call  and,  in  the  case  illustrated,  the  final  trunk  group  to  Indianapolis 
may  be  quite  large. 

The  method  employed  by  the  crossbar  tandem  marker  in  selecting 
the  direct  route  and  subsequent  alternate  routes  is  shown  in  simplified 
form  on  Fig.  5.  As  a  result  of  the  translating  operation,  the  marker 
selects  the  first  choice  route  relay,  corresponding  to  the  called  destina- 
tion. Each  route  relay  has  a  number  of  contacts  which  are  connected  to 
supply  all  the  information  recjuired  for  proper  routing  of  the  call.  Several 
of  these  contacts  are  used  to  indicate  the  equipment  location  of  the 
trunks  and  the  number  of  trunks  to  be  tested.  The  marker  tests  all  of 
the  trunks  in  the  direct  route  and  if  they  are  busy,  the  search  for  an 
idle  trunk  continues  in  the  first  alternate  route  which  is  brought  into 
play  from  the  "route  advance"  cross-connection  shown  on  the  sketch. 
As  many  as  three  alternate  routes  in  addition  to  the  first  choice  route 
can  be  tested  in  this  manner. 

STORING  AND  SENDING  FORWARD  DIGITS  AS  REQUIRED 

The  crossbar  tandem  equipment  at  control  switching  points  must 
store  all  the  digits  received  and  send  forward  as  many  as  are  required  to 
complete  the  call. 

The  called  number  recorded  at  a  switching  point  is  in  the  form  of 
ABX-XXXX  if  the  call  is  to  be  completed  in  the  same  numbering 
plan  area.  If  the  called  destination  is  in  another  area,  the  area  code 
XOX  or  XIX  precedes  the  7  digit  number.  The  area  codes  XOX  or  XIX 
and  the  local  office  code  ABX  are  the  digits  used  for  routing  purposes 
and  are  sufficient  to  complete  the  call  regardless  of  the  number  of  switch- 
ing points  involved.  Each  control  switching  point  is  arranged  to  ad- 
vance the  call  towards  its  destination  when  these  codes  are  received. 
If  the  next  switching  point  is  not  in  the  numbering  area  of  the  called 
telephone,  the  complete  ten-digit  number  is  needed  to  advance  the 
call  toward  its  destination.  If  the  next  switching  point  is  in  the  num- 


100  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 

bering  area  of  the  called  telephone  the  area  code  is  not  needed  and  seven 
digits  will  suffice  for  completing  the  call. 

For  example,  suppose  a  call  is  originated  by  a  customer  in  South 
Bend,  Indiana,  destined  for  customer  NAtional  4-1234  in  Washington, 
D.C.  If  it  is  assumed  that  the  route  to  Washington  is  via  a  switching 
center  in  Pittsburgh,  then  the  crossbar  tandem  equipment  at  South 
Bend  pulses  forward  to  Pittsburgh  202-NA4-1234,  202  being  the  area 
code  for  the  District  of  Columbia.  Pittsburgh  in  turn  will  delete  the 
area  code  and  send  NA4-1234  to  the  District  of  Columbia  terminating 
area. 

As  another  example,  suppose  the  crossbar  tandem  office  at  South 
Bend  receives  a  call  from  some  foreign  area  destined  to  a  nearby  step- 
by-step  end  office  in  Michigan.  The  crossbar  tandem  equipment  re- 
ceives and  stores  a  ten-digit  number  comprising  the  area  code  and  the- 
seven  digits  for  the  office  code  and  station  number.  Assuming  that 
direct  trunks  to  the  step-by-step  end  office  in  Michigan  are  available, 
the  area  code  and  office  code  are  deleted  and  the  line  number  only  is 
pulsed  forward.  To  meet  all  conditions,  the  equipment  is  arranged  to 
permit  deletion  of  either  the  first  three,  four,  five  or  six  digits  of  a  ten- 
digit  number. 

CODE  CONVERSION 

At  the  present  time,  some  step-by-step  primary  centers  reach  other 
offices  by  the  use  of  routing  codes  that  are  different  from  those  assigned 
under  the  national  numbering  plan.  This  arrangement  is  used  to  obtain 
economies  in  switching  equipment  of  the  step-by-step  plant  and  is 
accetpable  with  operator  originated  calls.  However,  with  the  intro- 
duction of  customer  direct  distance  dialing,  it  is  essential  that  the  codes 
used  by  customers  be  in  accordance  with  the  national  numbering  plan. 
The  crossbar  tandem  control  switching  point  must  then  automatically 
provide  the  routing  codes  needed  by  the  intermediate  step-by-step 
primary  centers.  This  is  accomplished  by  the  code  conversion  feature 
which  substitutes  the  arbitrary  digits  required  to  reach  the  called  office 
through  the  step-by-step  systems.  Fig.  6  illustrates  an  application  of 
this  feature.  It  shows  a  crossbar  tandem  office  arranged  for  completing 
calls  through  a  step-by-step  toll  center  to  a  local  central  office,  GArden 
8,  in  an  adjacent  area.  A  call  reaching  the  crossbar  tandem  office  for  a 
customer  in  this  office  arrives  with  the  national  number,  218-GA8-1234. 
To  complete  this  call,  the  crossbar  tandem  equipment  deletes  the  area 
code  218  and  pulses  forward  the  local  office  code  and  number.  If  the 


« 


CROSSBAK   TANDEM   AS    A   TOLL    SWITCHING    SYSTEM 


101 


call  is  switched  to  an  alternate  route  via  the  step-by-step  primary 
center,  it  will  be  necessary  for  the  crossbar  tandem  equipment  to  delete 
the  area  code  218  and  substitute  the  arbitrary  digits  062  to  direct  the 
call  through  the  switches  at  the  primary  center,  since  the  toll  center 
requires  the  full  seven  digit  number  for  completing  the  call. 

PREFIXING  DIGITS 

It  may  be  necessary  to  route  a  call  from  one  area  to  another  and  back 
to  the  original  area  for  completion.  Such  a  situation  arises  on  a  call 
from  Amarillo  to  Lubbock,  Texas,  both  in  area  915  when  the  crossbar 
tandem  switching  equipment  finds  all  of  the  direct  paths  from  Amarillo 
to  Lubbock  busy  as  illustrated  on  Fig.  7.  The  call  could  be  routed  to 
Lubbock  via  Oklahoma  City  which  is  in  area  405.  A  seven-digit  number 
for  example,  MAin  2-1234,  is  received  in  the  crossbar  tandem  office  at 
Amarillo.  Assuming  that  the  call  is  to  be  switched  out  of  the  915  area 
through  the  405  area  and  back  to  the  915  area  for  completion,  it  is 
necessary  for  the  crossbar  tandem  office  in  Amarillo  to  prefix  915  to  the 
MAin  2-1234  number  so  that  the  switching  equipment  in  Oklahoma 
City  will  know  that  the  call  is  for  the  915  area  and  not  for  the  405  area. 

Prefixing  digits  may  also  be  needed  at  crossbar  tandem  offices  to 
route  calls  through  step-by-step  primary  centers.  The  crossbar  tandem 
office  in  Fig.  8  receives  the  seven  digit  number  MA2-1234  for  a  call  to  a 


701 

AREA 


218 

AREA 


NUMBER 

RECEIVED 

218-GA8-1234 


CROSSBAR 
TANDEM 


NUMBER 

OUTPULSED 

062-GA8-1234 


STEP-BY-STEP 
PRIMARY  CENTER 


ALTERNATE 
ROUTE 

DIRECT 
ROUTE 


GA8-I234 


i^ 


GA8-t234 


SX  S 

TOLL 
CENTER 


GA8-1234 


LOCAL 
CUSTOMER 


Fig.  6  —  Code  conversion. 


102 


THE    BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 


customer  in  the  Madison  office  in  the  same  area.  However,  since  the 
toll  center  needs  the  full  seven  digit  number  for  completing  the  call  and 
since  the  step-by-step  switches  at  the  primary  center  "use  up"  two 
digits  (04)  for  its  switching,  the  crossbar  tandem  equipment  must 
prefix  04  to  the  seven  digit  number. 


METHOD  OF  DETERMINING  DIGITS  TO  BE  TRANSMITTED 

The  circuitry  involved  for  transmitting  digits  as  received,  prefixing, 
code  conversion  and  for  deletion  involves  both  marker  and  sender 
functions.  The  senders  have  ten  registers  (1  to  10)  for  storing  incoming 
digits  and  three  registers  (A A,  AB,  AC)  for  storing  the  arbitrary  digits 
that  are  used  for  prefixing  and  code  conversion. 

On  a  ten-digit  call  into  a  crossbar  tandem  switchmg  center  the  area 
code  XOX,  the  office  code  ABX  and  the  station  number  XXXX  are 
stored  in  the  inpulsing  or  receiving  registers  of  the  sender.  The  code 
digits  XOX-ABX  are  sent  to  the  marker  which  translates  them  to 
determine  which  of  the  digits  received  by  the  sender  should  be  outpulsed. 
It  also  determines  whether  arbitrary  digits  should  be  transmitted  ahead 
of  the  digits  received  and,  if  so,  the  value  of  the  arbitrary  digits  to  be 
stored  in  the  sender  registers  AA,  AB  and  AC.  Case  1  of  Fig.  9  assumes 
that  a  ten-digit  number  has  been  stored  in  the  sender  registers  1  to  10 


915 

AREA 


INCOMING 
TOLL  CALL 


LOCAL 

OFFICE 


AMARILLO 
CROSSBAR 

TANDEM 
OFFICE 


NUMBER 

RECEIVED 

MA2-1234 


405 
AREA 


^< 

■^ 

.-^ 

^^o^^ 

.-^^^"J^^' 

OKLAHOMA  CITY 
TOLL  OFFICE 

LUBBOCK 

TOLL 

OFFICE 

MA  2 

LOCAL 

CO. 

CUSTOMER 

MA  2-1234 


Fig.  7  —  Prefixing. 


CROSSBAR   TANDEM   AS   A   TOLL   SWITCHING   SYSTEM 


103 


and  that  the  marker  has  mformed  the  sender  the  called  number  is  to  be 
sent  as  received.  The  outpulsing  control  circuit  is  connected  to  each 
register  in  turn  through  the  steering  circuit  SI,  S2,  etc.  and  sends  the 
digits  stored. 

Case  2  illustrates  a  situation  where  the  sender  has  stored  ten  digits 
in  registers  1  to  10  and  received  information  from  the  marker  to  delete 
the  digits  in  registers  1  to  3  inclusive  and  to  substitute  the  arbitrary 
digits  stored  in  registers  AA,  AB  and  AC.  The  outpulsing  circuit  is 
first  connected  to  register  AA  through  steering  circuit  PSl,  then  to  AB 
through  PS2,  continuing  in  a  left  to  right  sequence  until  all  digits  are 
outpulsed. 

Case  3  covers  a  condition  where  the  sender  has  stored  seven  digits  and 
has  obtained  information  from  the  marker  to  prefix  the  two  digits 
stored  in  registers  AB  and  AC.  Outpulsing  begins  at  the  AB  register 
through  steering  circuit  PS2  and  then  advances  through  steering  circuit 
PS3  to  the  AC  register,  continuing  in  a  left  to  right  seciuence  until  all 
digits  have  been  transmitted. 

These  are  only  a  few  of  the  many  combinations  that  are  used  to  give 
the  crossbar  tandem  control  switching  equipment  complete  pulsing 
flexibility. 


SIX-DIGIT  TRANSLATION 

Six-digit  translation  will  be  another  feature  added  to  the  crossbar 
tandem  system.  When  only  three  digits  are  translated,  it  is  necessary  to 
direct  all  calls  to  a  foreign  area  over  a  single  route.  The  ability  to  trans- 
late six  digits  permits  the  establishment  of  two  or  more  routes  from  the 
switching  center  to  or  towards  the  foreign  area.  This  is  shown  in  Fig. 


LOCAL 
OFFICE 


NUMBER  OUTPULSED 
04-MA2-1234 


MADISON 
OFFICE 


MA2-I234 


CROSSBAR 
TANDEM 

t 

■     » 

' 

' — 1  n 

MA2- 
1 

1234 

EIVED 
4 



4 

1 
1 

TOLL 
CENTER 

— >- 

MADISON   2- 

1234 


STEP-BY-STEP 
PRIMARY  CENTER 


Fig.  8  —  Prefixing. 


104  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 


f/l 


10  with  Madison  and  Milwaukee,  Wisconsin,  in  area  414  and  Belle 
Plaine  Crossbar  Tandem  in  Chicago,  Illinois,  in  area  312.  An  economical 
trunking  plan  may  provide  for  direct  circuits  from  Chicago  to  each 
place.  If  only  three-digit  translation  were  provided  in  the  Chicago 
switching  equipment,  the  route  to  both  places  would  be  selected  as  a 
result  of  the  translation  of  the  414  area  code  alone  and,  therefore,  calls 
to  central  offices  reached  through  Madison,  would  need  to  be  routed 
via  Milwaukee.  This  involves  not  only  the  extra  trunk  mileage,  ])ut 
also  the  use  of  an  extra  switching  point.  With  six-digit  translation,  both 
the  area  code  and  the  central  office  code  are  analyzed,  making  it 
possible  to  select  the  direct  route  to  either  city. 

Six-digit  translation  in  crossbar  tandem  will  involve  primarily  the 
use  of  a  foreign  area  translator  and  a  marker.  The  translator  will  have 
a  capacity  for  translation  of  five  foreign  areas  and  for  60  routes  to  each 
area.  Since  the  translator  holding  time  is  very  short,  one  translator  is 
sufficient  to  handle  all  of  the  calls  requiring  six-digit  translation,  but 
two  are  always  provided  for  hazard  and  maintenance  reasons. 

On  a  call  requiring  six-digit  translation  the  first  three  digits  are 


CASE    1                ^ 

DIGITS   RECEIVED 

t 

2 

3 

-IMPULSING 
4             5 

REGISTERS - 
6            7 

8 

9 

10 

\ 

X 

0 

X 

A 

B 

X 

X 

X 

X 

X 

. 

. 

; 

i. 

OUTPULSING 
CONTROL 

;Si 

:  Sd 

S  J 

Sa     - 

S3 

So 

.  S  / 

-  So 

b»      *  oiu 

CASE    2 

DIGITS    RECEIVED 


OUTPULSING 
CONTROL 


0 


DIGITS    CODE    CONVERTED 
AA         AB         AC 


;:  PS1 


X' 


PS2   ;:PS3   ;:S4     ):S6     ~:S6     ;;S7     ::S8 


10 


59     ;:S10 


CASE  3 

DIGITS    RECEIVED 


DIGITS    PREFIXED 
AB         AC 


B' 


C 


OUTPULSING 
CONTROL 


i  PS2   : :  P 


B 


PS3     •:SI       ':S2      ::S3     :;S4     : :  S5      :'S6      ■;S7 


Fig.  9  —  Method  used  for  outpulsing  digits. 


CROSSBAR   TANDEM   AS   A   TOLL   SWITCHING   SYSTEM 


105 


translated  in  the  marker  and  the  second  three  digits  in  a  foreign  area 
translator  which  is  associated  with  the  marker.  Fig.  11  shows,  in  simpli- 
fied form,  how  this  translation  is  accomplished. 

The  first  three  digits,  corresponding  to  the  area  code,  are  received  by 
a  relay  code  tree  in  the  marker  which  translates  it  into  one  of  a  thousand 
code  points.  This  code  point  is  cross-connected  to  the  particular  relay  of 
the  five  area  relays  A(3-A4  which  has  been  assigned  to  the  called  area. 
A  foreign  area  translator  is  now  connected  to  the  marker  and  a  corre- 
sponding area  relay  is  operated  in  it.  The  translator  also  receives  the 
called  office  code  from  the  sender  via  the  marker  and  by  means  of  a 
relay  code  tree  similar  to  that  in  the  marker  translates  the  office  code 
to  one  of  a  thousand  code  points.  This  code  point  plus  the  area  relay  is 
sufficient  to  determine  the  actual  route  to  be  used.  As  shown  on  the 
sketch,  wires  from  each  of  the  code  points  are  threaded  through  trans- 
formers, two  for  each  area.  When  the  marker  is  ready  to  receive  the 
route  information,  a  surge  of  current  is  sent  through  one  of  these  threaded 
wires  which  produces  a  voltage  in  the  output  winding  to  ionize  the 
T-  and  U-  tubes.  Only  the  tubes  associated  with  the  area  involved  in 
the  translation  pass  current  to  operate  one  each  of  the  eight  T-  and  U- 
relays.  This  information  is  passed  to  the  marker  and  registered  on 
corresponding  tens  and  units  relays.  These  operate  a  route  relay  which 


WISCONSIN 


MICH. 


J 


ILLINOIS 


CHICAGO  = 
'  f BELLE  \ 

1     AREA        IplaINeJ 
\312  I 

^- — 1 


I  N  D. 


ROUTE    WITHOUT   6   DIGIT   TRANSLATION 


ROUTE   WITH   6   DIGIT    TRANSLATION 


Fig.  10  —  Six-digit  translation. 


106  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 


Fig.  11  —  Method  used  for  foreign  area  translation. 


CROSSBAR   TANDEM   AS   A   TOLL   SWITCHING   SYSTEM  107 

provides  all  the  information  necessary  for  routing  the  call  to  the  central 
office  involved. 

CUSTOMER  DIRECT  DISTANCE  DIALING 

Crossbar  tandem  will  provide  arrangements  permitting  customers  in 
step-by-step  offices  to  dial  their  own  calls  anywhere  in  the  country. 
Centralized  automatic  message  accounting  previously  mentioned  will 
be  used  for  charging  purposes.  While  the  basic  plan  for  direct  distance 
dialing  provides  for  the  dialing  of  either  seven  or  ten  digits,  it  will  be 
necessary  for  the  customer  in  step-by-step  areas  to  prefix  a  three-digit 
directing  code,  such  as  112,  to  the  called  number.  This  directing  code 
is  required  to  direct  the  call  through  the  step-by-step  switches  to  the 
crossbar  tandem  office  so  that  the  seven  or  ten  digit  number  can  be 
registered  in  the  crossbar  tandem  office. 

When  a  customer  in  a  step-by-step  office  originates  a  call  to  a  distant 
customer  whose  national  number  is  915-CH3-1234,  he  first  dials  the 
directing  code  112  and  then  the  ten-digit  number.  The  dialing  of  112 
causes  the  selectors  in  the  step-by-step  office  to  select  an  outgoing  trunk 
to  the  crossbar  tandem  office.  The  incoming  trunk  in  the  crossbar  tandem 
office  has  quick  access  to  a  three-digit  register.  The  register  must  be 
connected  during  the  interval  between  the  last  digit  of  the  directing 
code  and  the  first  digit  of  the  national  number  to  insure  registration  of 
this  number.  This  arrangement  is  used  to  permit  the  customer  to  dial 
all  digits  without  delay  and  avoids  the  use  of  a  second  dial  tone.  If  this 
arrangement  were  not  used,  the  customer  would  be  required  to  wait 
after  dialing  the  112  until  the  trunk  in  the  tandem  crossbar  office  could 
gain  access  to  a  sender  through  the  sender  link  circuit  which  would 
then  signal  the  customer  to  resume  dialing  by  returning  dial  tone. 

After  recording  the  915  area  code  digits  in  the  case  assumed,  the 
CH3-1234  portion  of  the  number  is  registered  directly  in  the  tandem 
sender  which  has  been  connected  to  the  trunk  while  the  customer  was 
dialing  915.  When  the  sender  is  attached  to  the  trunk,  it  signals  the 
three-digit  register  to  transfer  the  915  area  code  digits  to  it  via  a  con- 
nector circuit.  Thus  when  dialing  is  complete,  the  entire  number  915- 
CH3-1234  is  registered  in  the  sender. 

Crossbar  tandem  is  being  arranged  to  serve  customers  of  panel  and 
No.  1  crossbar  offices  for  direct  distance  dialing.  At  the  present  time, 
ten  digit  direct  distance  dialing  is  not  available  to  these  customers 
because  the  digit  storing  equipments  in  these  offices  are  limited  to 
eight  digits.  Developments  now  under  way,  will  provide  arrangements 
for  expanding  the  digit  capacity  in  the  local  offices  so  that  ultirnately 


108  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

calls  from  custoniers  in  panel  and  No.  1  crossbar  offices  may  be  routed 
through  crossbar  tandem  cr  other  equivalent  offices  to  telephones 
anywhere  in  the  country. 

CONCLUSION 

The  new  features  developed  for  crossbar  tandem  will  adapt  it  to 
switching  all  types  of  traffic  at  many  important  switching  centers  of 
the  nationwide  toll  network.  Of  the  225  important  toll  switching  centers 
now  contemplated,  it  is  expected  that  about  80  of  these  will  be  ecjuipped 
with  crossbar  tandem. 

REFERENCES 

1.  Collis,  R.  E.,  Crossbar  Tandem  System,  A.I.E.E.  Trans.,  69,  pp.  997-1004, 1950. 

2.  King,  G.  v..  Centralized  Automatic  Message  Accounting,  B.S.T.J.,  33,  pp. 

1331-1342,  1952. 

3.  Nunn,  W.  H.,  Nationwide  Numbering  Plan,  B.S.T.J.,  31,  pp.  851-859,  1952. 

4.  Pilliod,  J.  J.,  Fundamental  Plans  for  Toll  Telephone  Plant,  B. S.T.J. ,  31,  pp. 

832-850,  1952. 

5.  Shipley,  F.  F.,  Automatic  Toll  Switching  Systems,  B.S.T.J.,  31,  pp.  860-882, 

1952. 

6.  Truitt,  C.  J.,  Traffic  Engineering  Techniques  for  Determining  Trunk  Require- 

ments in  Alternate  Routing  Trunk  Networks,  B.S.T.J.,  33,  pp.  277-302,  1954. 

7.  Clos,  C,  Automatic  Alternate  Routing  of  Telephone  Traffic,  Bell  Laboratories 

Record,  32,  pp.  51-57,  Feb.  1954. 


Growing  Waves  Due  to  Transverse 

Velocities 

By  J.  R.  PIERCE  and  L.  R.  WALKER 

(Manuscript  received  March  30,  1955) 

This  paper  treats  propagation  of  slow  waves  in  two-dimensional  neu- 
tralized electron  floiv  in  which  all  electrons  have  the  same  velocity  in  the 
direction  of  propagation  hut  in  which  there  are  streams  of  two  or  more  veloci- 
ties normal  to  the  direction  of  propagation.  In  a  finite  beam  in  which 
'  electrons  are  reflected  elastically  at  the  boundaries  and  in  which  equal  dc 
currents  are  carried  by  electrons  with  transverse  velocities  -\-Ui  and  —  Wi  , 
there  is  an  antisi/mmetrical  growing  ivave  if 

Up   ~  {rUi/Wf 

and  a  symmetrical  growing  wave  if 


y- 


i{Tu,/wy 


Here  cop  is  plasma  frequency  for  the  total  charge  density  and  W  is  beam 
width. 

INTKODUCTION 

i  It  is  well-known  that  there  can  be  growing  waves  in  electron  flow  when 
the  flow  is  composed  of  several  streams  of  electrons  having  different 
velocities  in  the  direction  of  propagation  of  the  waves.  '  While  Birdsall 
considers  the  case  of  growing  waves  in  electron  flow  consisting  of  streams 
which  cross  one  another,  the  growing  waves  which  he  finds  apparently 
occur  when  two  streams  have  different  components  of  velocity  in  the 
direction  of  propagation. 

This  paper  shows  that  there  can  be  growing  waves  in  electron  flow 
consisting  of  two  or  more  streams  with  the  same  component  of  velocity 
in  the  direction  of  wave  propagation  but  with  different  components  of 
velocity  transverse  to  the  direction  of  propagation.  Such  growing  Avaves 
can  exist  when  the  electric  field  varies  in  strength  across  the  flow.  Such 
waves  could  result  in  the  amplification  of  noise  fluctuations  in  electron 

'  flow.  They  could  also  be  used  to  amplify  signals. 

109 


110  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

Actual  electron  flow  as  it  occurs  in  practical  tubes  can  exhibit  trans- 
verse velocities.  For  instance,  in  Brillouin  flow,  '  •  if  we  consider  electron 
motion  in  a  coordinate  system  rotating  with  the  Larmor  frequency  we 
see  that  electrons  with  transverse  velocities  are  free  to  cross  the  beam 
repeatedly,  being  reflected  at  the  boundaries  of  the  beam.  The  trans- 
verse \-elocities  may  be  completely  disorganized  thermal  velocities,  or 
they  may  be  larger  and  better-organized  velocities  due  to  aberrations  at 
the  edges  of  the  cathode  or  at  lenses  or  apertures.  Two-dimensional 
Brillouin  flow  allows  similar  transverse  motions. 

It  would  be  difficult  to  treat  the  case  of  Brillouin  or  Brillouin-like  flow 
with  transverse  velocities.  Here,  simpler  cases  with  transverse  velocities 
will  be  considered.  The  first  case  treated  is  that  of  infinite  ion-neutra- 
lized two-dimensional  flow  with  transverse  velocities.  The  second  case 
treated  is  that  of  two-dimensional  flow  in  a  beam  of  finite  width  in  which 
the  electrons  are  elastically  reflected  at  the  boundaries  of  the  beam. 
Growing  waves  are  found  in  both  cases,  and  the  rate  of  growth  may  be 
large. 

In  the  case  of  the  finite  beam  both  an  antisymmetric  mode  and  a 
symmetric  mode  are  possible.  Here,  it  appears,  the  current  density 
required  for  a  growing  wave  in  the  symmetric  mode  is  about  ^^  times 
as  great  as  the  current  density  required  for  a  growing  wa^•e  in  the  anti- 
symmetric mode.  Hence,  as  the  current  is  increased,  the  first  growing 
waves  to  arise  might  be  antisymmetric  modes,  which  could  couple  to  a 
symmetrical  resonator  or  helix  only  through  a  lack  of  symmetry  or 
through  high-level  effects. 

1 .  Infinite  two-dimensional  flow 

Consider  a  two-dimensional  problem  in  which  the  potential  varies 
sinusoidally  in  the  y  direction,  as  exp{—j^z)  in  the  z  direction  and  as  exp 
(jut)  with  time.  Let  there  be  two  electron  streams,  each  of  a  negative 
charge  po  and  each  moving  with  the  velocity  ?/o  in  the  z  direction,  but 
with  velocities  Wi  and  —ih  respectively  in  the  y  direction.  Let  us  denote 
ac  quantities  pertaining  to  the  first  stream  by  subscripts  1  and  ac  quan- 
tities pertaining  to  the  second  stream  by  subscripts  2.  The  ac  charge 
density  will  be  denoted  by  p,  the  ac  velocity  in  the  y  direction  by  y, 
and  the  ac  velocity  in  the  z  direction  by  i.  We  will  use  linearized  or 
small-signal  equations  of  motion.^  We  will  denote  differentiation  with 
respect  to  ?/  by  the  operator  D. 

The  equation  of  continuity  gives 

jupi  =   -D(piUi  +  po?yi)  +  j|8(piWo  +  pnii)  (1.1)1 

jcopo  =    -D{-p-iHi  -\-  pi)lj':d  +  il3(P2''o  +  Poi2)  (1.2) 


t; 


GROWING   WAVES    DUE   TO    TRANSVERSE    VELOCITIES  111 

Let  US  define 

dx  =  i(co  -  ^u,)  +  u,D  (1.3) 

do  =  ./(w  -  i8wo)  -  uj)  (1.4) 

We  can  then  rewrite  (1.1)  and  (1.2)  as 

f/iPi  =  Poi-Diji  +  j(3zi)  (1.5) 

dopi  =  Pi^{  —  Dy2  +  .7/3i2)  (1.0) 

We  will  assume  that  we  are  dealing  ^^•ith  slow  waves  and  can  use  a  po- 
tential V  to  describe  the  field.  We  can  thus  write  the  linearized  equations 
of  motion  in  the  form 

r/iii  =   -j-^F  (1.7) 

m 

d2h  =   -j-^V  (1.8) 

m 

drlji  =  -  DV  (1.9) 

m 

d,y,  =  1  DV  (1.10) 

w 

From  (1.5)  to  (1.10)  we  obtain 

^m  =  ~  PoiD'  -  ^')V  (1.11) 

m 

d'p2=  --poiD'-  ^')V  (1.12) 

m 

Now,  Poisson's  equation  is 

{D'  -  ^')V  =  _^L±£!  (1.13) 

From  (1.11)  to  (1.13)  we  obtain 

{D'  -  /3^)y  =  -   Kco/  (^1  +  ^^  (D'  -  /3^)7  (1.14) 


9    ^ 
—  Z—  po 

2  m 

Wp     =  

e 

Here  Wp  is  the  plasma  frequency  for  the  charge  of  both  beams. 


(1.15) 


112  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 


Either 


or  else 


(2)'  -  /3')7  =  0 

—  C0„"   (c/l"    +   ^2") 


^  2         di^  d.} 

We  will  consider  this  second  case. 

W(<  should  note  from  (1.3)  and  (1.4)  that 

d{  =  u^-D^  -  (co  -  /5(/„)"  +  2yD(co  -  |8?/.o)«i 

^2^  =  ?<i-D"  -  (co  -  ^ihf  -  2jD{o^  -  l3uo)ui 

di'  +  f/o'  =  2{u{D'  -  (co  -  iSwo)'] 

rfiW  =  [uiD'  +  (co  -  /3;/„)T 

Thus,  (1.17)  becomes 


(1.16) 


(1.17) 


(1.18) 
(1.19) 
(1.20) 
(1.21) 

(1.22) 
WD""  +  (co  -  j8mo)2]^ 

If  the  quantities  involved  vary  sinusoidally  with  y  as  cos  ru  or  sin  yy, 


-co, 


\u{lf  -  (co  -  /3ao)'] 


then 


Our  equation  becomes 


D' 


-7 


(1.23) 


CO 


P       L 


1    + 


CO  —  jS'Uo 


T^Wi^ 


_  /co  -  13^0 Y" 
\      7^1      / 


(1.24) 


What  happens  if  we  have  many  transverse  velocities?  If  we  refer  back 
to  (1.14)  we  see  that  we  will  have  an  equation  of  the  form 


1  =  E  -  14 


2^pn 


2  I  din     +   C?2n 


d^d     ^      J  ^^-^''^ 

"In     (fin        / 

Here  cop„^  is  a  plasma  frequency  based  on  the  density  of  electrons  having 
transverse  velocities  ±Un  .  Equation  (1.25)  can  be  written 

(co  -  |(3//o)""| 


i  =  E 


A^ 

'M„2  r    _  (g,  -  /3uo)2-['^ 

L  7-'"n^  J 


(1.2()) 


GROWING   WAVES   DUE   TO    TRANSVERSE    VELOCITIES 


113 


(u;-/3Uo 


Fig.  1 

Suppose  we  plot  the  left-hand  and  the  right-hand  sides  of  (1.26)  versus 
(co  —  ^Uo)-  The  general  appearance  of  the  left-hand  and  right-hand  sides 
of  (1.26)  is  indicated  in  Fig.  1  for  the  case  of  two  velocities  Un  .  There 
will  always  be  two  unattenuated  waves  at  values  of  (w  —  /3wo)  >  y  Ug 
where  Ue  is  the  extreme  value  of  lu;  these  correspond  to  intersections  3 
and  3'  in  Fig.  2.  The  other  waves,  two  per  value  of  Un  ,  may  be  unat- 
tenuated or  a  pair  of  increasing  and  decreasing  waves,  depending  on  the 
values  of  the  parameters.  If 


CO 


pn 


-yhir? 


>  1 


there  will  be  at  least  one  pair  of  increasing  and  decreasing  waves. 

It  is  not  clear  what  will  happen  for  a  Maxwellian  distribution  of  veloci- 
ties. However,  we  must  remember  that  various  aberrations  might  give  a 
very  different,  strongly  peaked  velocity  distribution. 

Let  us  consider  the  amount  of  gain  in  the  case  of  one  pair  of  transverse 
velocities,  ±i/i  .  The  equation  is  now 


2      2 
7  Ui 

C0„2 


[ 


1    + 


CO  —  |3wo 


)•] 


[  ■  -  (^OI 


(1.27) 


Let 


/5  =  ^+i^ 

Wo  Wo 


(1.28) 


114  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


1  .u 
0.9 
0.8 

\ 

\ 

0.7 
0.6 

\ 

\ 

\, 

\ 

^ 

0.5 
0.4 
0.3 

\^ 

\ 

>s. 

\ 

V 

0.2 

\ 

> 

\ 

0.1 

\ 

\ 

0 

\ 

0  0.1         0.2         0.3         0.4        0.5         0.6        0.7         0.8        0.9         1.0 

v2 


m 

Fig.  2 
This  relation  defines  e.  Equation  (1.27)  becomes 


2       2 
0}J 


1  -  e^ 


(1  +  e^)^  ^'-''^ 

In  Fig.  2,  e  is  plotted  versus  the  parameter  y^Ui/oip^.  We  see  that  as  the 
parameter  falls  below  unity,  e  increases,  at  first  rapidly,  and  then  more 
slowly,  reaching  a  value  of  ±1  as  the  parameter  goes  to  zero  (as  cop' 
goes  to  infinity,  for  instance). 

It  will  be  shown  in  Section  2  of  this  paper  that  these  results  for  infinite 
flow  are  in  some  degree  an  approximation  to  the  results  for  flow  in  narrow 
beams.  It  is  therefore  of  interest  to  see  what  results  they  yield  if  applied 
to  a  beam  of  finite  width. 

If  the  beam  has  a  length  L,  the  voltage  gain  is 


The  gain  G  in  db  is 


G  =  8.7  '^  €  db 

Wo 


(1.30) 


(1.31) 


GROWING   WAVES   DUE   TO   TRANSVERSE   VELOCITIES  115 

Let  the  width  of  the  beam  be  W.  We  let 

Thus,  for  n  =   1,  there  is  a  half -cycle  variation  across  the  beam.  From 
(1.31)  and  (1.32) 

G  =  27.s(^^^\ne  db  (1.33) 


Now  L/uo  is  the  time  it  takes  the  electrons  to  go  from  one  end  of  the 
beam  to  the  other,  while  W/ui  is  the  time  it  takes  the  electrons  to  cross 
the  beam.  If  the  electrons  cross  the  beam  A''  times 

iV  =  ^4  (1-34) 

Thus, 

G  =  27.SNnedb  (1.35) 

While  for  a  given  value  of  e  the  gain  is  higher  if  we  make  the  phase 
vary  many  times  across  the  beam,  i.e.,  if  we  make  n  large,  we  should 
note  that  to  get  any  gain  at  all  we  must  have 


2    .        //iTTUlV 
0)r>     > 


(1.36) 


W 


If  we  increase  oop  ,  which  is  proportional  to  current  density,  so  that  cop 
passes  through  this  value,  the  gain  will  rise  sharply  just  after  cOp"  passes 
through  this  value  and  will  rise  less  rapidly  thereafter. 

.?.  A  Two-Dimensional  Beam  of  Finite  Width. 

Let  us  assume  a  beam  of  finite  width  in  the  ^/-direction ;  the  boundaries 
lying  a,t  y  =  ±^o  •  It  will  be  assumed  also  that  electrons  incident  upon 
these  boundaries  are  elastically  reflected,  so  that  electrons  of  the  incident 
stream  (1  or  2)  are  converted  into  those  of  the  other  stream  (2  or  1).  The 
condition  of  elastic  reflection  implies  that 

yi  =  -h  (2.1) 

Zi  =  22    Sit  y  =  ±2/0  (2.2) 

and,  in  addition,  that 

Pi  =  p2    at  y  =  ±?/o  .  (2.3) 

since  there  is  no  change  in  the  number  of  electrons  at  the  boundary. 


116  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

The  equations  of  motion  and  of  continuity  (1.7-1.12)  may  be  satisfied 
by  introducing  a  single  quantity,  ^,  such  that 

V  =  dx  dzV  (2.4) 

ii  =  -J  -  /3  d,  ^2^  (2.5) 

m 

zi  =  —j  —  di  di\p  (2.6) 

m 

yi=-d,  d^Dyp  (2.7) 

m 

112=-  di  d^Di^  (2.8) 

m 


Pi 


m 


poiD'  -  ^')  dirl^  (2.9) 


P2  =  --  Po(i)'  -  n  di'rl^  (2.10) 

m 


Then,  if  we  introduce  the  symbol,  12,  for  co  —  jSuo 

yi  +  y^  =  2j-d,d2D^yp  (2.11)  ' 

m 

h-  Z2  =  2j  -  di  diUiD^  (2.12) 

m 

PI  -  P2  =  2j-  po{D'  -  l3')uiQDi^  (2.13) 

m 

It  is  clear  that  if 

Drjy  =  D^xl^  =  0        y  =  ±yo  (2.14) 

the  conditions  for  elastic  reflection  will  be  satisfied.  The  equation  satis- 
fied by  rf/  may  now  be  found  from  Poisson's  equation,  (1-13),  and  is 

{D'  -  /3^)  dx'  di^P  =  '-^{D'-  fi'){d,'  +  di)^l. 

we 

or 

{D'  -  ^')[{u,'D'  +  ny  +  coJiu.'D'  -  n')]  =  0  (2.15) 

which  is  of  the  sixth  degree  in  D.  So  far  four  boundary  conditions  have, 
been  imposed.  The  remaining  necessary  pair  arise  from  matching  the 


GROWING   WAVES    DUE   TO   TRANSVERSE    VELOCITIES  117 

internal  fields  to  the  external  ones.  For  y  >  ijo 

V  =  Voe-'^'-e~^"  (2.16) 


and 


Similarlv 


^  +  i37  =  0         at  2/  =  2/0 
dy 


dV 

—  -  ^V  =  0        at  y  =  -7/0  (2.17) 

dy 

The  most  familiar  procedure  now  would  be  to  look  for  solutions  of 
(2,15)  of  the  form,  e''^.  This  would  give  the  sextic  for  c 

(c'  -  /3')[(WiV  +  nY  +  a;/(niV  -  n')]   =  0  (2.18) 

with  the  roots  c  =  ±|8,  ±ci  ,  ±C2  ,  let  us  s^y.  We  could  then  express  \p 
as  a  linear  combination  of  these  six  solutions  and  adjust  the  coefficients 
to  satisfy  the  six  boundary  equations.  In  this  way  a  characteristic  equa- 
tion for  l3  would  be  obtained.  From  the  S3anmetry  of  the  problem  this 
has  the  general  form  F(l3,  Ci)  =  F(i3,  C2),  where  Ci  and  Co  are  found  from 
;  (2.18).  The  discussion  of  the  problem  in  these  terms  is  rather  laborious 
and,  if  we  are  concerned  mainly  with  examining  qualitatively  the  onset 
of  increasing  waves,  another  approach  serves  better. 

From  the  symmetry  of  the  equations  and  of  the  boundary  conditions 
we  see  that  there  are  solutions  for  \p  (and  consequently  for  V  and  p) 
which  are  even  in  y  and  again  some  which  are  odd  in  y.  Consider  first  the 
even  solutions.  We  will  assume  that  there  is  an  even  function,  ^i(y), 
periodic  in  y  with  period  2yo ,  which  coincides  with  \l/(y)  in  the  open 
interval,  —yo<y<yo  and  that  \pi(:y)  has  a  Fourier  cosine  series  repre- 
sentation : 

hiy)  =  E  c„  cos  \ny        X„  =  —        n  =  0, 1,  2,  •  •  •       (2.19) 
1  yo 

yp  inside  the  interval  satisfies  (2.15),  so  we  assume  that  ypiiy)  obeys 
(D^  -  ^')[{u,'D'  +  ^'f  +  o.,\u,'D'  -  ^-)^, 


+00 


(2.20) 


=    Z)   5(2/  -  2m  +  lyo) 


where  6  is  the  familiar  5-function.  Since  D\p  and  D^\p  are  required  to  vanish 
at  the  ends  of  the  interval  and  \l/,  D'^  and  Z)V  are  even  it  follows  that  all 


118  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

of  these  functions  are  continuous.  We  assume  that  xpi  =  \l/,  D\pi  =  D\l/, 
DVi  =  D~\p,  D%  =  D^yp  and  D%  =  D*xl/  at  the  ends  of  the  intervals. 
From  (2.20),  Wi'D^i  ^  -H  as  y  ^  ijo  . 
Since 

2  8iy  -  2m  +  lyo)  =  ^  +  -  £  (-1)"  cos  Ky        (2.21) 

we  obtain  from  (2.20) 
/  1 


2?/oi/'i 


,/32ff(i22    -    Wp2) 


+  2i;(-l)"  ^"^'"^ 


Since 

^  +  ^F  =  (Z)  +  /3)(t.x^Z)^  +  fi^)V, 

using  (2.4),  the  condition  for  matching  to  the  external  field, 

dV 

^  +  /37  =  0, 

dy 

yields,  using  D\p  =  DV  =  0  and  Ui*D^\f/  =  —  i^,  the  relation 

(ui'D'  +  fi')Vi  =  3^/3     at  2/  =  2/0  . 
Applying  this  to  (2.22),  we  then  obtain,  finally, 
yo  ^  1 


+  2Z 


r  (^2   4-    X„2)[(i22    -    Ml2X„2)2    -    cOp2(Q2    +    ,,^2X„2)] 


(2.22) 


(2.23) 


For  the  odd  solution  we  use  a  function,  yp2(y),  equal  to  ;/'(?/)  in  — //o  < 
y  <  yo  and  representable  by  a  sine  series.  To  ensure  the  vanishing  of  D^p 
and  7)V  at  ?/  =  ±?/o  it  is  appropriate  to  use  the  functions,  sin  n„y,  where 
Mn  =  (n  -\-  l'2)ir/yo  .  The  period  is  now  iyo  and  we  define  \p2(y)  in  /yo  < 
y  <  32/0  by  the  relation  i;'2(2/)  =  ^{2yo  —  y)  and  in  —  32/o  <  2/  <  —  2/o  by 
^2(2/)  =  ^{  —  '^Uo  —  y)-  Thus,  we  write 

00 

1^2(2/)    =    2  C?n  sin  UnV  Hn    =    (w  +   3^)^7/0 

0 

^2(2/)  ^^i"  ho  supposed  to  satisfy 


GROWING   "WAVES   DUE   TO   TRANSVERSE   VELOCITIES  119 

+M  (2.24) 

=    2   [^(y  -  4m  +  lyo)  -  Ky  -  4m  -  lyo)] 

m=— 00 

The  extended  definition  of  i/'2  (outside  — /yo  <y  <  ijo)  is  such  that  we  may 

again  take  \pi  =  \p, ,  D%  =  DV  at  the  ends  of  the  interval.  ?/i*DVi  is 

still  equal  to  —  }4  at  ij  =  ijo .  Now 

+  00  

£  [5(y  -  4m  +  iW  -  ^(y  —  4m  -  l^/o)] 

(2.25) 

=  —  2  (—1)"  sin  /i„?/ 
2/0 

so  from  (2.24)  we  may  find 

v^L    =  -T (-l)"sin/xnj/ ,        ^ 

Matching  to  the  external  field  as  before  gives 
and  applied  to  (2.26)  we  have 

00  /rfi  2      2\2 

_y^  =  y (^  -  uinn) ,     . 

The  equations  (2.23)  and  (2.27)  for  the  even  and  odd  modes  may  be 
rewritten  using  the  following  reduced  variables. 

.  =  ^« 

IT 
1     _   Wj/0   _   Wo 

(2.23)  becomes 

^'       4-  2  y  ^ (n'  -  k^  _   _  . 

and  (2.27)  transforms  to 


„^  2^  +  (n  +  3^)2  [{n  +  1^)2  -  /c2]2  -  s\{n  +  3^)^  +  k']         (2  99) 

=    — tt;? 


120 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 


We  shall  assume  in  considering  (2.28)  and  (2.29)  that  the  beam  is 
sufficiently  wide  for  the  transit  of  an  electron  from  one  side  to  the  other 
to  take  a  few  RF  cycles.  The  number  of  cycles  is  in  fact,  coz/o/ttwi  ,  and, 
hence,  from  the  definition  of  z,  we  see  that  for  values  of  A:  less  than  2, 
perhaps,  z  is  certainly  positive. 

Let  us  consider  (2.29)  first  since  it  proves  to  be  the  simpler  case.  If  we 
transfer  the  term  ttz  to  the  right  hand  side,  it  follo^^•s  from  the  observa- 
tion that  z  is  positive  (for  modest  values  of  h),  that  it  is  necessary  to 
make  the  sum  negative.  The  sum  may  be  studied  qualitatively  by  sketch- 
ing in  the  k^  —  d'  plane  the  lines  on  which  the  individual  terms  go  to 
infinity,  given  by 

[(n  +  3^)^  -  k'f 


8'  = 


(n  -f  K)'  +  k' 


(2.30) 


3.5 


Fig.  3 


GROWING   WAVES    DUE   TO   TRANSVERSE   VELOCITIES 


121 


77 


0.4 

0.3 


0.2         0.4         0.6  0.8  1.0 


1.2  1.4 

(X/TT 


1.6 


1.8         2.0         2.2        2.4 


Fig.  4 

Fig.  3  shows  a  few  such  curves  (n  =  0,  1,  2).  To  the  right  of  such  curves 
the  individual  term  in  question  is  negative,  except  on  the  Hne,  k^  = 
{n  +  V^)  ,  where  it  attains  the  value  of  zero.  Approaching  the  curves 
from  the  right  the  terms  go  to  —  oo .  On  the  left  of  the  curves  the  func- 
tion is  positive  and  goes  to  +  oo  as  the  curve  is  approached  from  the 


10 


... 

/ 

/ 

/ 

/ 

J, 

L 

/ 

/ 

/ 

/ 

/ 

/ 

L 

/ 

/ 

/ 

/ 

Y 

/ 

/=, 

/ 

A 

V 

/ 

/ 

/ 

' 

\ 

/ 

A 

-0 

1 

/ 

\ 

/ 

y 

/ 

\ 

^^ 

A 

>< 

■^ 

^^ 

>C 

^ 

"\ 

^ 

'^ 

^ 

3  4  5  6  7  8  9 

Fig.  5 


J  22  THE   BELL   SYSTEM   TECHNICAL  JOURNAL,    JANUARY    195G 

left.  Clearly  in  the  regions  marked  +  which  lie  to  the  left  of  every  curve 
given  by  (2.30),  the  sum  is  positive  and  we  cannot  have  roots.  Let  us 
examine  the  sum  in  the  region  to  the  right  of  the  n  =  0  curve  and  to  the 
left  of  all  others.  On  the  line,  A;^  =  J4»  the  sum  is  positive,  since  the  first 
term  is  zero.  On  any  other  line,  k'  =  constant,  the  sum  goes  from  +  °° 
at  the  n  =  1  curve  monotonically  to  —  oo  at  the  n  =  0  curve,  so  that 
somewhere  it  must  pass  through  0.  This  enables  us  to  draw  the  zero- 
sum  contours  qualitatively  in  this  region  and  they  are  indicated  in  Fig.  3. 
We  are  now  in  a  position  to  follow  the  variation  in  the  sum  as  k  varies 
at  fixed  5  .  It  is  readily  seen  that  for  5  <  0.25,  because  —wz  is  negative 
in  the  region  under  consideration,  there  will  be  four  real  roots,  tw^o  for 
positive,  two  for  negative  k.  For  5'  slightly  greater  than  0.25,  the  sum  has 


Fig.  6A 


GROWING   WAVES   DUE   TO   TRANSVERSE   VELOCITIES  123 

a  deep  minimum  for  k  =  0,  so  that  there  are  still  four  real  roots  unless  z 
is  very  large.  For  z  fixed,  as  5^  increases,  the  depth  of  the  minimum  de- 
creases and  there  will  finally  occur  a  5"  for  which  the  minimum  is  so  shal- 
low that  two  of  the  real  roots  disappear.  Call  z(0)  the  value  of  ziork  =  0, 
write  the  sum  as  2(5^  k^)  and  suppose  that  2(5o^  0)  =  —irziO),  then  for 
small  k  we  have 

S(5^  e)  =  -«(0)  +  (6^  -  8o')  §,  +  k'§,=  -«(0)  -"^  k 

do^  dk^  Ua 

as 

dB  dk^ 


^  =  ^±        /     ".^(^-^0^)  + 


'^     a/ 
dk'     y 

The  roots  become  complex  when 


aA-2 


S.2  J  2  (Ul/Uo) 

0    =  do    — 


52  as 

d8^  dB 


Since  Ui/uq  may  be  considered  small  (say  10  per  cent)  it  is  sufficient  to 


look  for  the  values  of  5o^. 
When  k   =  0  we  have 


-TZ  =  2X) 

2z 


z  (n  +  y,y 


z^  +  52 

irz" 


z'-\-in-\-  y^r  (n  -1-  y^y-  -  s' 

'  H ^ + i ^ 

0    \in  +  3^)2  -  52  ^  (n  +  1^)2  +  zy 
(5  tan  -Kb  -\-  z  tanh  irz) 


z"  +  52 


Fig.  4  shows  the  solution  of  this  equation  for  various  2(0)  or  oiyo/iruo . 
Clearly  the  threshold  5  is  rather  insensitive  to  variations  in  uyo/ir^io . 

Equation  (2.28)  may  be  examined  by  a  similar  method,  but  here  some 
complications  arise.  Fig.  5  shows  the  infinity  curves  for  n  =  0,  1,  2,  3; 
the  n  =  0  term  being  of  the  form  k^/k^  —  8^.  The  lowest  critical  region 
in  5^  is  the  neighborhood  of  the  point  fc^  =  6^  =  ]^i,  which  is  the  intersec- 
tion of  the  n  =  0  and  n  =  1  lines.  To  obtain  an  idea  of  the  behavior  of 


124  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    195G 

the  left  hand  side  (l.h.s.)  of  (2.28)  in  this  area  we  first  see  how  the  point 
k^  =  f  =  1^  can  be  approached  so  that  the  l.h.s.  remains  finite.  If  we 
put  k^  =  H  +  £  and  a'  =  ^  +  ce  and  expand  the  first  two  dominant 
terms  of  (2.28),  then  adjust  c  to  keep  the  result  finite  as  f  -^  0  we  find 

=  1  3^'  -  5 
^  ~  4  32^  +  1 

c  varies  from  —  %  to  \i  as  z  goes  from  0  to  c»  ,  changing  sign  at  2^  =  %. 
Every  curve  for  which  the  l.h.s.  is  constant  makes  quadratic  contact  with 
the  Jine  5"  —  V3  =  c(/v"  —  ]i)  at  Jc'  =  5'  =  I/3.  If  we  remember  that 
the  l.h.s.  is  positive  for  A;'  =  0,  0  <  5"  <  1  and  for  A;^  =  1,  0  <  5^  <  1, 


1 

2 

lik 

3 

w-oX 

k^ 

/ 

1 

y(  I 

3 

SHADED    AREAS            // 
NEGATIVE              yV 

X  /' 
/  // 

/  /I 
/  /  / 

X 

\ 

n  =  i^v 

0 

3 


3 


Fig.  6B 


GROWING   WAVES   DUE   TO   TRANSVERSE   VELOCITIES  125 

since  there  are  no  negative  terms  in  the  sum  for  these  ranges  and  again 
that  the  l.h.s.  must  change  sign  between  the  n  =  0  and  n  —  I  Unes  for 
any  k^  in  the  range  0  <  k^  <  1  (since  it  varies  from  T  oo  to  ±0°),  this 
information  may  be  combined  with  that  about  the  immediate  vicinity 
of  5  =  k  =  V^  to  enable  us  to  draw  a  Hue  on  which  the  l.h.s.  is  zero. 
This  is  indicated  in  Figs.  6A  and  6B  for  small  z  and  large  z  respec- 
tively. It  will  be  seen  that  the  zero  curve  and,  in  fact,  all  curves  on  which 
the  l.h.s.  is  equal  to  a  negative  constant  are  required  to  have  a  vertical 
tangent  at  some  point.  This  point  may  be  above  or  below  /c^  =  ^  (de- 
pending upon  the  sign  of  c  or  the  size  of  z)  but  always  at  a  3^  >  ^.  For 
5  <  H  there  are  no  regions  where  roots  can  arise  as  we  can  readily  see 
by  considering  how  the  l.h.s.  varies  with  k"^  at  fixed  5^  For  a  fixed  d^  >  }/s 
we  have,  then,  either  for  k^  >  ]4  or  k^  <  V^,  according  to  the  size  of  z, 
a  negative  minimum  which  becomes  indefinitely  deep  as  5^  -^  ^.  Thus, 
since  the  negative  terms  on  the  right-hand  side  are  not  sensitive  to  small 
changes  in  5^,  we  must  expect  to  find,  for  a  fixed  value  of  the  l.h.s.,  two 
real  solutions  of  (2.28)  for  some  values  of  5^  and  no  real  solutions  for  some 
larger  value  of  5  ,  since  the  negative  minimum  of  the  l.h.s.  may  be  made 
as  shallow  as  we  like  by  increasing  6".  By  continuity  then  we  expect  to 
find  pairs  of  complex  roots  in  this  region.  Rather  oddly  these  roots,  which 
will  exist  certainly  for  5'  sufficiently  close  to  V^  +  0,  will  disappear  if 
5^  is  sufficiently  increased. 

REFERENCES 

1.  L.  S.  Nergaard,  Analysis  of  a  Simple  Model  of  a  Two-Beam  Growing-Wave 

Tube,  RCA  Review,  9,  pp.  585-601,  Dec,  1948. 

2.  J.  R.  Pierce  and  W.  B.  Hebenstreit,  A  New  Type  of  High-Frequency  Amplifier, 

B.  S.  T.  J.,  28,  pp.  23-51,  Jan.,  1949. 

3.  A.  V.  Haeff,  The  Electron-Wave  Tube  —  A  Novel  Method  of  Generation  and 

Amplification  of  Microwave  Energy,  Proc.  I.R.E.,  37,  pp.  4-10,  Jan.,  1949. 

4.  G.  G.  Macfarlg,ne  and  H.  G.  Hay,  Wave  Propagation  in  a  Slipping  Stream  of 

Electrons,  Proc.  Physical  Society  Sec.  B,  63,  pp.  409-427,  June,  1950. 

5.  P.  Gurnard  and  H.  Huber,  Etude  E.xp^rimentale  de  L'Interaction  par  Ondes 

de  Chargd^d'Espace  au  Sein  d'Un  Faisceau  Electronique  se  Deplagant  dans 
Des  Champs  Electrique  et  Magn^tique  Croisfe,  Annales  de  Radio^lectricite, 
7,  pp.  252-278,  Oct.,  1952. 

6.  C.  K.  Birdsall,  Double  Stream  Amplification  Due  to  Interaction  Between  Two 

Oblique  Electron  Streams,  Technical  Report  No.  24,  Electronics  Research 
Laboratory,  Stanford  University. 

7.  L.  Brillouin,  A  Theorem  of  Larmor  and  Its  Importance  for  Electrons  in  Mag- 

netic Fields,  Phys.  Rev.,  67,  pp.  260-266,  1945. 

8.  J.  R.  Pierce,  Theory  and  Design  of  Electron  Beams,  2nd  Ed.,  Chapter  9,  Van 

Nostrand,  1954. 

9.  J.  R.  Pierce,  Traveling-Wave  Tubes,  Van  Nostrand,  1950. 


Coupled  Helices 

By  J.  S.  COOK,  R.  KOMPFNER  and  C.  F.  QUATE 

(Received  September  21,  1955) 

An  analysis  of  coupled  helices  is  presented,  using  the  transmission  line 
approach  and  also  the  field  approach,  with  the  objective  of  providing  the 
tube  designer  and  the  microwave  circuit  engineer  with  a  basis  for  approxi- 
mate calcidations.  Devices  based  on  the  presence  of  only  one  mode  of  propa- 
gation are  briefly  described;  and  methods  for  establishing  such  a  mode  are 
given.  Devices  depending  on  the  simultaneous  presence  of  both  modes,  that 
is,  depending  on  the  beat  wave  phenomenon,  are  described;  some  experi- 
mental results  are  cited  in  support  of  the  view  that  a  novel  and  useful  class  of 
coupling  elements  has  been  discovered. 

CONTENTS 

1.  Introduction 129 

2.  Theory  of  Coupled  Helices 132 

2.1  Introduction 132 

2.2  Transmission  Line  Equations 133 

2.3  Solution  for  Synchronous  Helices 135 

2.4  Non-Synchronous  Helix  Solutions 137 

2.5  A  Look  at  the  Fields 139 

2.6  A  Simple  Estimate  of  b  and  x 141 

2.7  Strength  of  Coupling  versus  Frequency 142 

2.8  Field  Solutions 144 

.  2.9  Bifilar  Helix 146 

2.10  Effect  of  Dielectric  Material  between  Helices 148 

2.11  The  Conditions  for  Maximum  Power  Transfer 151 

2.12  Mode  Impedance 152 

3.  Applications  of  Coupled  Helices 154 

3.1  Excitation  of  Pure  Modes 156 

3.1.1  Direct  Excitation 156 

3.1.2  Tapered  Coupler 157 

3.1.3  Stepped  Coupler 158 

3.2  Low  Noise  Transverse  Field  Amplifier 159 

3.3  Dispersive  Traveling  Wave  Tube 159 

3.4  Devices  Using  Both  Modes 161 

3.4.1  Coupled  Helix  Transducer 161 

3.4.2  Coupled-Helix  Attenuator 165 

4.  Conclusion 167 

Appendix 

I    Solution  of  Field  Equations 168 

II    Finding  r I73 

III    Complete  Power  Transfer 175 

127 


128  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

GLOSSARY   OF   SYMBOLS 

a  Mean  radius  of  inner  helix 

h  Mean  radius  of  outer  helix 

h  Capacitive  coupling  coefficient 

Bio,  20     shunt  susceptance  of  inner  and  outer  helices,  respectively 

Bi,  2       Shunt  susceptance  plus  mutual  susceptance  of  inner  and  outer 

helices,  respectively,  Bm  +  Bm  ,  Boo  +  B^ 
Bm         Mutual  susceptance  of  two  coupled  helices 
c  Velocity  of  light  in  free  space 

d  Radial  separation  between  helices,  h-a 

D  Directivity  of  helix  coupler 

E  Electric  field  intensity 

F  Maximum  fraction  of  power  transferable  from  one  coupled  helix 

to  the  other 
F(ya)     Impedance  parameter 

7i,  2        RF  current  in  inner  and  outer  helix,  respectively 
K  Impedance  in  terms  of  longitudinal  electric  field  on  helix  axis 

and  axial  power  flow 
L  ]\Iinimum  axial  distance  required  for  maximum  energy  transfer 

from  one  coupled  helix  to  the  other,  X6/2 

Axial  power  flow  along  helix  circuit 

Radial  coordinate 

Radius  where  longitudinal  component  of  electric  field  is  zero  for 

transverse  mode  (about  midway  between  a  and  b) 

Return  loss 

Radial  separation  betw^een  helix  and  adjacent  conducting  shield 

Time 

RF  potential  of  inner  and  outer  helices,  respectively  • 

Inductive  coupling  coefficient 

Series  reactance  of  inner  and  outer  helices,  respectively 

Series  reactance  plus  mutual  reactance  of  inner  and  outer  helices, 

respectively,  Xio  +  Xm  ,  X20  +  Xm 

Mutual  reactance  of  two  coupled  helices 

Axial  coordinate 

Impedance  of  inner  and  outer  helix,  respectively 

Attenuation  constant  of  inner  and  outer  helices,  respectively 

General  circuit  phase  constant;  or  mean  circuit  phase  constant. 

Free  space  phase  constant 

Axial  phase  constant  of  inner  and  outer  helices  in  absence  of 

coupling,  V^ioXio ,  VBioXio 


p 

r 
f 

R 

s 
t 

F1.2 

X 

Xva,  20 
Xl,  2 

Xm 

Z 
Zil,  2 

Oil,  2 

^0 
^10.  20 

COUPLED   HELICES  129 

181 , 2  May  be  considered  as  axial  phase  constant  of  inner  and  outer 
helices,  respectively 

(Sft  Beat  phase  constant 

jSc  Coupling  phase  constant,  (identical  with  ^b  when  /3i  =  JS2) 

I3ce  Coupling  phase  constant  when  there  is  dielectric  material  be- 

tween the  helices 

/3d  Difference  phase  constant,  [  /3i  —  /32  [ 

(8f  Axial  phase  constant  of  single  helix  in  presence  of  dielectric 

^t,  (  Axial  phase  constant  of  transverse  and  longitudinal  modes,  re- 
spectively 

7  Radial  phase  constant 

jt,  (  Radial  phase  constant  of  transverse  and  longitudinal  modes, 
respectively 

r  Axial  propagation  constant 

Tt.  (  Axial  propagation  constant  for  transverse  and  longitudinal 
coupled-helix  modes,  respectively 

e  Dielectric  constant 

e'  Relative  dielectric  constant,  e/eq 

En  Dielectric  constant  of  free  space 

X  General  circuit  wavelength;  or  mean  circuit  wavelength,  \/XiX2 

Xo  Free  space  wavelength 

Xi,  2        Axial  wavelength  on  inner  and  outer  helix,  respectively 

X6  Beat  wavelength 

Xc  Coupling  wavelength  (identical  with  Xb  when  (5i  =  /So) 

yj/  Helix  pitch  angle 

i/'i,  2        Pitch  angle  of  inner  and  outer  helix,  respectively 

CO  Angular  frequency 

1.    INTRODUCTION 

Since  their  first  appearance,  traveling-wave  tubes  have  changed  only 
very  little.  In  particular,  if  we  divide  the  tube,  somewhat  arbitrarily, 
into  circuit  and  beam,  the  most  widely  used  circuit  is  still  the  helix,  and 
the  most  widely  used  transition  from  the  circuits  outside  the  tube  to  the 
circuit  inside  is  from  waveguide  to  a  short  stub  or  antenna  which,  in 
turn,  is  attached  to  the  helix,  either  directly  or  through  a  few  turns  of 
increased  pitch.  Feedback  of  signal  energy  along  the  helix  is  prevented 
by  means  of  loss,  either  distributed  along  the  whole  helix  or  localized 
somewhere  near  the  middle.  The  helix  is  most  often  supported  along  its 
whole  length  by  glass  or  ceramic  rods,  which  also  serve  to  carry  a  con- 
ducting coating  ("aquadag"),  acting  as  the  localized  loss. 

We  therefore  find  the  following  circuit  elements  within  the  tube  en- 
velope, fixed  and  inaccessible  once  and  for  all  after  it  has  been  sealed  off: 


130  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

1 .  The  helix  itself,  determining  the  beam  voltage  for  optimum  beam- 
circuit  interaction ; 

2.  The  helix  ends  and  matching  stubs,  etc.,  all  of  which  have  to  be 
positioned  very  precisely  with  relation  to  the  waveguide  circuits  in 
order  to  obtain  a  reproducible  match ; 

3.  The  loss,  in  the  form  of  "aquadag"  on  the  support  rods,  which 
greatly  influences  the  tube  performance  by  its  position  and  distril)ution. 

In  spite  of  the  enormous  bandwidth  over  which  the  traveling-wave 
tube  is  potentially  capable  of  operating  —  a  feature  new  in  the  field  of 
microwave  amplifier  tubes  —  it  turns  out  that  the  positioning  of  the  tube 
in  the  external  circuits  and  the  necessary  matching  adjustments  are 
rather  critical;  moreover  the  overall  bandwidths  achieved  are  far  short 
of  the  obtainable  maximum. 

Another  fact,  experimentally  observed  and  well-founded  in  theory, 
rounds  off  the  situation:  The  electro-magnetic  field  surrounding  a  helix, 
i.e.,  the  slow  wave,  under  normal  conditions,  does  not  radiate,  and  is 
confined  to  the  close  vicinity  of  the  helix,  falling  off  in  intensity  nearly 
exponentially  with  distance  from  the  helix.  A  typical  traveling-wave 
tube,  in  which  the  helix  is  supported  by  ceramic  rods,  and  the  whole 
enclosed  by  the  glass  envelope,  is  thus  practically  inaccessible  as  far  as 
RF  fields  are  concerned,  with  the  exception  of  the  ends  of  the  helix, 
where  provision  is  made  for  matching  to  the  outside  circuits.  Placing 
objects  such  as  conductors,  dielectrics  or  distributed  loss  close  to  the 
tube  is,  in  general,  observed  to  have  no  effect  whatsoever. 

In  the  course  of  an  experimental  investigation  into  the  propagation  of 
space  charge  waves  in  electron  beams  it  was  desired  to  couple  into  a  long 
helix  at  any  point  chosen  along  its  length.  Because  of  the  feebleness  of 
the  RF  fields  outside  the  helix  surrounded  by  the  conventional  sup- 
ports and  the  envelope,  this  seemed  a  rather  difficult  task.  Nevertheless, 
if  accomplished,  such  a  coupling  would  have  other  and  even  more  im- 
portant applications;  and  a  good  deal  of  thought  was  given  to  the 
problem. 

Coupled  concentric  helices  were  found  to  provide  the  solution  to  the 
problem  of  coupling  into  and  out  of  a  helix  at  any  particular  point,  and  to 
a  number  of  other  problems  too. 

Concentric  coupled  helices  have  been  considered  by  J.  R.  Pierce, 
who  has  ti'cated  the  problem  mainly  with  transverse  fields  in  mind. 
Such  fields  were  thought  to  be  useful  in  low-noise  traveling-wave  tube 
devices.  Pierce's  analysis  treats  the  helices  as  transmission  lines  coupled 
uniformly  over  their  length  by  means  of  nuitual  distributed  capacitance 
and  inductance.  Pierce  also  recognized  that  it  is  necessary  to  wind  the 


COUPLED   HELICES  l,']! 

two  helices  in  opposite  directions  in  order  to  obtain  well  defined  trans- 
verse and  axial  wave  modes  which  are  well  separated  in  respect  to  their 
velocities  of  propagation. 

Pierce  did  not  then  give  an  estimate  of  the  velocity  separation  which 
might  be  attainable  with  practical  helices,  nor  did  anybody  (as  far  as  we 
are  aware)  then  know  how  strong  a  coupling  one  might  obtain  with  such 
heUces. 

It  was,  therefore,  a  considerable  (and  gratifying)  surprise^'  ^  to  find 
that  concentric  helices  of  practically  realizable  dimensions  and  separa- 
tions are,  indeed,  very  strongly  coupled  when,  and  these  are  the  im- 
portant points, 

(a)  They  have  very  nearly  equal  velocities  of  propagation  when  un- 
coupled, and  when 

(b)  They  are  wound  in  opposite  senses. 

It  was  found  that  virtually  complete  power  transfer  from  outer  to 
inner  helix  (or  vice  versa)  could  be  effected  over  a  distance  of  the  order 
of  one  helix  wavelength  (normally  between  i^fo  and  3^^o  of  a  free-space 
wavelength. 

It  was  also  found  that  it  was  possible  to  make  a  transition  from  a  co- 
axial transmission  line  to  a  short  (outer)  helix  and  thence  through  the 
glass  surrounding  an  inner  helix,  which  was  fairly  good  over  quite  a  con- 
siderable bandwidth.  Such  a  transition  also  acted  as  a  directional  coupler, 
RF  power  coming  from  the  coaxial  line  being  transferred  to  the  inner 
helix  predominantly  in  one  direction. 

Thus,  one  of  the  shortcomings  of  the  "conventional"  helix  traveling- 
wave  tube,  namely  the  necessary  built-in  accuracy  of  the  matching 
parameters,  was  overcome  by  means  of  the  new  type  of  coupler  that 
might  evolve  around  coupled  helix-to-helix  systems. 

Other  constructional  and  functional  possibilities  appeared  as  the 
work  progressed,  such  as  coupled-helix  attenuators,  various  tj^pes  of 
broadband  couplers,  and  schemes  for  exciting  pure  transverse  (slow)  or 
longitudinal  (fast)  waves  on  coupled  helices. 

One  central  fact  emerged  from  all  these  considerations:  by  placing 
part  of  the  circuit  outside  the  tube  envelope  with  complete  independence 
from  the  helix  terminations  inside  the  tube,  coupled  helices  give  back  to 
the  circuit  designer  a  freedom  comparable  only  with  that  obtained  at 
much  lower  frequencies.  For  example,  it  now  appears  entirely  possible 
to  make  one  type  of  traveling  wave  tube  to  cover  a  variety  of  frequency 
bands,  each  band  requiring  merely  different  couplers  or  outside  helices, 
the  tube  itself  remaining  unchanged. 

Moreover,  one  tube  may  now  be  made  to  fulfill  a  number  of  different 


132  THE    BELL   SYSTEM   TECHNICAI-   JOURNAL,    JANUARY    1956 

functions;  this  is  made  possible  by  the  freedom  with  which  couplers 
and  attenuators  can  be  placed  at  any  chosen  point  along  the  tube. 

Considerable  work  in  this  field  has  been  done  elsewhere.  Reference 
will  be  made  to  it  wherever  possible.  However,  only  that  work  with 
which  the  authors  have  been  intimately  connected  will  be  fully  reported 
here.  In  particular,  the  effect  of  the  electron  beam  on  the  wave  propaga- 
tion phenomena  will  not  be  considered. 

2.    THEORY   OF   COUPLED   HELICES 

2.1  Introduction 

In  the  past,  considerable  success  has  been  attained  in  the  under- 
standing of  traveling  wave  tube  behavior  by  means  of  the  so-called 
"transmission-line"  approach  to  the  theory.  In  particular,  J.  R,  Pierce 
used  it  in  his  initial  analysis  and  was  thus  able  to  present  the  solution 
of  the  so-called  traveling-wave  tube  equations  in  the  form  of  4  waves, 
one  of  which  is  an  exponentially  growing  forward  traveling  wave  basic 
to  the  operation  of  the  tube  as  an  amplifier. 

This  transmission-line  approach  considers  the  helix  —  or  any  slow- 
wave  circuit  for  that  matter  —  as  a  transmission  line  with  distributed 
capacitance  and  inductance  with  which  an  electron  beam  interacts. 
As  the  first  approximation,  the  beam  is  assumed  to  be  moving  in  an  RF 
field  of  uniform  intensity  across  the  beam. 

In  this  way  very  simple  expressions  for  the  coupling  parameter  and 
gain,  etc.,  are  obtained,  which  give  one  a  good  appreciation  of  the 
physically  relevant  quantities. 

A  number  of  factors,  such  as  the  effect  of  space  charge,  the  non-uniform 
distribution  of  the  electric  field,  the  variation  of  circuit  impedance  with 
frequency,  etc.,  can,  in  principle,  be  calculated  and  their  effects  can  be 
superimposed,  so  to  speak,  on  the  relatively  simple  expressions  deriving 
from  the  simple  transmission  line  theory.  This  has,  in  fact,  been  done  and 
is,  from  the  design  engineer's  point  of  view,  quite  satisfactory. 

However,  phj^sicists  are  bound  to  be  unhappy  over  this  state  of 
affairs.  In  the  beginning  was  Maxwell,  and  therefore  the  proper  point  to 
start  from  is  Maxwell. 

So-called  "Field"  theories  of  traveling-w^ave  tubes,  based  on  Maxwell's 
equation,  solved  with  the  appropriate  boundary  conditions,  have  been 
worked  out  and  their  main  importance  is  that  they  largely  confirm  the 
results  obtained  by  the  inexact  transmission  line  theory.  It  is,  however, 
in  the  nature  of  things  that  field  theories  cannot  give  answers  in  terms  of 


COUPLED    HELICES  133 

simple  closed  expressions  of  any  generality.  The  best  that  can  be  done 
is  in  the  form  of  curves,  with  step-wise  increases  of  particular  param- 
eters. These  can  be  of  considerable  value  in  particular  cases,  and  when 
exactness  is  essential. 

In  this  paper  we  shall  proceed  by  giving  the  "transmission-line"  type 
theory  first,  together  with  the  elaborations  that  are  necessary  to  arrive 
at  an  estimate  of  the  strength  of  coupling  possible  with  coaxial  helices. 
The  "field"  type  theory  will  be  used  whenever  the  other  theory  fails,  or 
is  inadequate.  Considerable  physical  insight  can  be  gotten  with  the  use 
of  the  transmission-line  theory;  nevertheless  recourse  to  field  theory  is 
necessary  in  a  number  of  cases,  as  will  be  seen. 

It  will  be  noted  that  in  all  the  calculations  to  be  presented  the  presence 
of  an  electron  beam  is  left  out  of  account.  This  is  done  for  two  reasons: 
Its  inclusion  would  enormously  complicate  the  theory,  and,  as  will 
eventually  be  shown,  it  would  modify  our  conclusions  only  very  slightly. 
Moreover,  in  practically  all  cases  which  we  shall  consider,  the  helices  are 
so  tightly  coupled  that  the  velocities  of  the  two  normal  modes  of  propaga- 
tion are  very  different,  as  will  be  shown.  Thus,  only  when  the  beam 
velocity  is  very  near  to  either  one  or  the  other  wave  velocity,  will 
growing-wave  interaction  take  place  between  the  beam  and  the  helices. 
In  this  case  conventional  traveling  wave  tube  theory  may  be  used. 

A  theory  of  coupled  helices  in  the  presence  of  an  electron  beam  has 
been  presented  by  Wade  and  Rynn,^  who  treated  the  case  of  weakly 
coupled  helices  and  arrived  at  conclusions  not  at  variance  with  our  views. 

2.2  Transmission  Line  Equations 

Following  Pierce  we  describe  two  lossless  helices  by  their  distributed 
series  reactances  Xio  and  A'20  and  their  distributed  shunt  susceptances 
Bio  and  ^20  .  Thus  their  phase  constants  are 

/3io  =  V^ioA'io 

Let  these  helices  be  coupled  by  means  of  a  mutual  distributed  reac- 
tance Xm  and  a  mutual  susceptance  B^  ,  both  of  which  are,  in  a  way 
which  will  be  described  later,  functions  of  the  geometry. 

Let  waves  in  the  coupled  system  be  described  by  the  factor 

jut    —  Tj; 

e    e 


\v 


here  the  F's  are  the  propagation  constants  to  be  found. 


134  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


The  transmission  line  equations  may  be  written: 

r/i  -  jB,V,  +  jB„y2  =  0 
rFi  -  iXi/i  +  jXJo  =  0 

r/o  -  JB0V2  +  jB„yi  =  0 

TV2    -  jXJa  +  jXJ,    -    0 

where 

B,  -  5io  +  5« 

Bo    =    B20   +    Bm 

X2    =    X20   -f"    Xm 

1 1  and  1 2  are  eliminated  from  the  (2.2.1)  and  we  find 

F2  ^  +  (r-  +  XiBi  +  x^Bj 

Fi 

F2 


(2.2.1) 


X\Bm   +  B%Xm 

+  (r-  +  X2S2  +  x^Bj 


XlBm    +    5lX„ 


(2.2.2) 


(2.2.3) 


These  two  equations  are  then  multipUed  together  and  an  expression  for 
r  of  the  4th  degree  is  obtained : 

r'  +  (XiBi  +  X2B2  +  2Z,„Bjr' 

+  (X1Z2  -  Xj){B,B2  -  Bj)  =  0 
We  now  define  a  number  of  dimensionless  quantities: 


(2.2.4) 


B, 


BiB. 


Xm 


=  h'  =  (eapacitive  coupling  coefficient)' 


=  X    =  (inductive  coupling  coefficient) 


XiXo 

B\Xi  =  ^1,        B2X2  =  (82' 
X1B1X2B2  =  13^  =  (mean  phase  constant) 
With  these  substitutions  we  obtain  the  general  equation  for  T~ 


T'  =  13' 


2  \(3-r  ^  I3{'  ^ 


y  4v^2'^^/3i^ 


_     (2.2.5) 


+  26.r      -  (1  -  .r-)(l  -  U') 


COUPLED    HELICES  135 


(2.2.6) 


If  we  make  the  same  substitutions  in  (2.2.2)  we  find 

Fi        T   ZiL    /3(/3i?>  + /3o:r)    . 
where  the  Z's  are  the  impedances  of  the  heUces,  i.e., 

Z,.  =  VXJB, 

2.3  Solution  for  Synchronous  Helices 

Let  us  consider  the  particular  case  where  (Si  =  (S-z  =  |S.  From  (2.2.5) 
we  obtain 

r'  =  -I3\l  +  xb  db  (x  +  b)]  (2.3.1) 

Each  of  the  above  values  of  T"  characterizes  a  normal  mode  of  propaga- 
tion involving  both  helices.  The  two  square  roots  of  each  T"  represent 
waves  going  in  the  positive  and  negative  directions.  We  shall  consider 
only  the  positive  roots  of  T  ,  denoted  Tt  and  Tt ,  which  represent  the 
forward  traveling  waves. 

Ttj  =  i/3Vl  +  xb  ±  {x  +  b)  (2.3.2) 

If  a:  >  0  and  6  >  0 

I  r, I  >  |/3i,  I  r,|  <  1^1 

Thus  Vt  represents  a  normal  mode  of  propagation  which  is  slower  than 
the  propagation  velocity  of  either  helix  alone  and  can  be  called  the 
"slow"  wave.  Similarly  T(  represents  a  "fast"  wave.  We  shall  find  that, 
in  fact,  X  and  b  are  numerically  equal  in  most  cases  of  interest  to  us;  we 
therefore  write  the  expressions  for  the  propagation  constants 

r.  =  M^  +  H(-^  +  b)] 

(2.3.3) 

r.  =  Ml  -  Viix  +  b)] 

If  we  substitute  (2.3.3)  into  (2.2.6)  for  the  case  where  /3i  =  (82  =  /3  and 
assume,  for  simplicity,  that  the  helix  self-impedances  are  equal,  we  find 
that  for  r  =  Tt 

Y%  _ 


for  r  =  T; 


F2 

-—  =  -f  1 

Yx       ^ 


136  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

Thus,  the  slow  wave  is  characterized  by  equal  voltages  of  unlike  sign  on 
the  two  helices,  and  the  fast  wave  by  equal  voltages  of  like  sign.  It  fol- 
lows that  the  electric  field  in  the  annular  region  between  two  such  coupled 
concentric  helices  will  be  transverse  for  the  slow  wave  and  longitudinal 
for  the  fast.  For  this  reason  the  slow  and  fast  modes  are  often  referred 
to  as  the  transverse  and  longitudinal  modes,  respectively,  as  indi- 
cated by  our  subscripts. 

It  should  be  noted  here  that  we  arbitrarily  chose  h  and  x  positive.  A 
different  choice  of  signs  cannot  alter  the  fact  that  the  transverse  mode  is 
the  slower  and  the  longitudinal  mode  is  the  faster  of  the  two. 

Apart  from  the  interest  in  the  separate  existence  of  the  fast  and  slow 
waves  as  such,  another  object  of  interest  is  the  phenomenon  of  the  simul- 
taneous existence  of  both  waves  and  the  interference,  or  spatial  beating, 
between  them. 

Let  V2  denote  the  voltage  on  the  outer  hehx;  and  let  Vi ,  the  voltage 
on  the  inner  halix,  be  zero  at  z  =  0.  Then  we  have,  omitting  the  common 
factor  e'"  , 

(2.3.4) 

Since  at  2  =  0,  Fi  =  0,  Vn  =  —  V(^  .  For  the  case  we  have  considered  we 
have  found  Fa  =  —  V^  and  Vn  =  V^  .  We  can  write  (2.3.4)  as 


Fi  =  I  {e~'^'  -  e-^n 


V,  =  ^  {e''^'  +  e-'n 


(2.3.5) 


F2  can  be  written 


=  Ye-"'''''^''^'''  cos  [-jj^(r,  -  Vi)z\ 
In  the  case  when  x  =  6,  and  /Si  =  /32  =  /8 

F2  =  Ye"'^'  cos  Wiix  +  h)^z\  (2.3.6) 

Correspondingly,  it  can  be  shown  that  the  voltage  on  the  inner  helix  is 

y,  =  j\Tfr^^'  sin  Wiix  +  h)^z\  (2.3.7) 

The  last  tAvo  equations  exhibit  clearly  what  we  have  called  the  spatial 
beat  phenomonou,  a  wave-like  transfer  of  power  from  one  helix  to  thc^ 


COUPLED   HELICES  .  137 

other  and  back.  We  started,  arbitrarily,  with  all  the  voltage  on  the  outer 
helix  at  2  =  0,  and  none  on  the  inner;  after  a  distance,  z',  which  makes 
the  argument  of  the  cosine  x/2,  there  is  no  voltage  on  the  outer  helix 
and  all  is  on  the  inner. 

To  conform  with  published  material  let  us  define  what  we  shall  call 
the  "coupling  phase-constant"  as 

^,  =  ^{h  +  x)  (2.3.8) 

From  (2.3.3)  we  find  that  for  (Si  =  ^2  =  |S,  and  x  =  h, 

Tt  -  Ti  =  jl3c 

2.4  Non-Synchronous  Helix  Solutions 

Let  us  now  go  back  to  the  more  general  case  where  the  propagation 
velocities  of  the  (uncoupled)  helices  are  not  equal.  Eciuation  (2.2.5)  can 
be  written: 


Further,  let  us  define 


(2.4.1) 


r-  =   -^-  [1  +  (1/2)A  +  xb  ± 

V(l  +  xb)A  +  (1/4)A2  +  (6  +  xy] 
where 

L       /3       _ 
In  the  case  where  x  =  h,  (2.4.1)  has  an  exact  root. 

r,, ,  =  j^  [Vl  +  A/4  ±  1/2  Va  +  (a;  +  by]  (2.4.2) 

We  shall  be  interested  in  the  difference  between  Tt  and  Tt, 

Tt-Tf  =  j^  Va  +  (x  +  by-  (2.4.3) 

Now  we  substitute  for  A  and  find 

Tt-  Tc  =  j  V(^i  -  ^2y  +  ^M&  +  4'  (2.4.4) 

Let  us  define  the  "beat  phase-constant"  as: 

Pb  =  V(/3i  -  /32)2  +  nb  +  xy 

so  that 

r,  -  r,  =  jA  (2.4.5) 


(3a  =  \  i5i  -  iSo 


138  THE    BELL    SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 

and  call  this  the  "difference  phase-constant,"  i.e.,  the  hase  constant  cor- 
responding to  two  uncoupled  waves  of  the  same  frequency  but  differing 
phase  velocities.  We  can  thus  state  the  relation  between  these  phase 
constants : 

^b'  =  &I  +  ^c  (2.4.6) 

This  relation  is  identical  (except  for  notation)  with  expression  (33)  in 
S.  E.  Miller's  paper. ^  In  this  paper  Miller  also  gives  expressions  for  the 
voltage  amplitudes  in  two  coupled  transmission  systems  in  the  case  of 
unequal  phase  velocities.  It  turns  out  that  in  such  a  case  the  power  trans- 
fer from  one  system  to  the  other  is  necessarily  incomplete.  This  is  of 
particular  interest  to  us,  in  connection  with  a  number  of  practical 
schemes.  In  our  notation  it  is  relatively  simple,  and  we  can  state  it  by 
saying  that  the  maximum  fraction  of  power  transferred  is 

(2.4.7) 


or,  in  more  detail, 


iS/  +  iSc-       (^1  -  iS2)-  +  ^Kh  +  xY 

This  relationship  can  be  shown  to  be  a  good  approximation  from  (2.2.6), 
(2.3.4),  (2.4.2),  on  the  assumption  that  h  ^  x  and  Zx  'PH  Z2 ,  and  the 
further  assumption  that  the  system  is  lossless;  that  is, 

I  72  I  ^  +  I  Fi  I  ^  =  constant  (2.4.8) 

We  note  that  the  phase  velocity  difference  gives  rise  to  two  phenomena : 
It  reduces  the  coupling  w^avelength  and  it  reduces  the  amount  of  power 
that  can  be  transferred  from  one  helix  to  the  other. 

Something  should  be  said  about  the  case  where  the  two  helix  imped- 
ances are  not  equal,  since  this,  indeed,  is  usually  the  case  with  coupled 
concentric  helices.  Equation  (2.4.8)  becomes: 

I  F2 1    _^  \Vx\_  ^  (3Qj^g^^j^^  (2.4.9) 


Z2  Z\ 

Using  this  relation  it  is  found  from  (2.3.4)  that 


F2  ,  /Zi 

FiT  z, 


(1  ±  Vl  -  /^)  (2.4.10) 


When  Ihis  is  combined  with  (2.2.6)  it  is  found  that  the  impedances  droj) 
out  with  the  voltages,  and  that  "F"  is  a  function  of  the  |S's  only.  In  other 


COUPLED   HELICES  139 

words,  complete  power  transfer  occurs  when  ,81  =  /So  regardless  of  the 
relative  impedances  of  the  helices. 

The  reader  will  remember  that  (3io  and  (820 ,  not  jSi  and  ^o ,  were  defined 
as  the  phase  constants  of  the  helices  in  the  absence  of  each  other.  If  the 
assumption  that  h  ^  x  is  maintained,  it  will  be  found  that  all  of  the  de- 
rived relationships  hold  true  when  (Sno  is  substituted  for  /3„  .  In  other 
words,  throughout  the  paper,  /3i  and  /So  may  be  treated  as  the  phase  con- 
stants of  the  inner  and  outer  helices,  respectively.  In  particular  it  should 
be  noted  that  if  these  ciuantities  are  to  be  measured  experimentally  each 
helix  must  be  kept  in  the  same  environment  as  if  the  helices  were  coupled ; 
onl}^  the  other  helix  may  be  removed.  That  is,  if  there  is  dielectric  in  the 
annular  region  between  the  coupled  helices,  /Si  and  ^2  must  each  be 
measured  in  the  presence  of  that  dielectric. 

Miller  also  has  treated  the  case  of  lossy  coupled  transmission  systems. 
The  expressions  are  lengthy  and  complicated  and  we  believe  that  no 
substantial  error  is  made  in  simply  applying  his  conclusions  to  our  case. 

If  the  attenuation  constants  ai  and  ao  of  the  two  transmission  systems 
(helices)  are  equal,  no  change  is  required  in  our  expressions;  when  they 
are  unequal  the  total  available  power  (in  both  helices)  is  most  effectively 
reduced  when 

^4^'^l  (2.4.11) 

Pc 

This  fact  may  be  made  use  of  in  designing  coupled  helix  attenuators. 

2.5  A  Look  at  the  Fields 

It  may  be  advantageous  to  consider  sketches  of  typical  field  distribu- 
tions in  coupled  helices,  as  in  Fig.  2.1,  before  we  go  on  to  derive  a  quanti- 
tative estimate  of  the  coupling  factors  actually  obtainable  in  practice. 

Fig.  2.1(a)  shows,  diagrammatically,  electric  field  lines  when  the 
coupled  helices  are  excited  in  the  fast  or  "longitudinal"  mode.  To  set  up 
this  mode  only,  one  has  to  supply  voltages  of  like  sign  and  equal  ampli- 
tudes to  both  helices.  For  this  reason,  this  mode  is  also  sometimes  called 
the  "(+-f)  mode." 

Fig.  2.1(b)  shows  the  electric  field  lines  when  the  helices  are  excited  in 
the  slow  or  "transverse"  mode.  This  is  the  kind  of  field  required  in  the 
transverse  interaction  type  of  traveling  wave  tube.  In  order  to  excite 
this  mode  it  is  necessary  to  supply  voltages  of  equal  amplitude  and 
opposite  signs  to  the  helices  and  for  this  reason  it  is  sometimes  called  the 
"(-| — )  mode."  One  way  of  exciting  this  mode  consists  in  connecting  one 


140  THE    BELL    SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

helix  to  one  of  the  two  conductors  of  a  balanced  transmission  line 
("Lecher"-line)  and  the  other  hehx  to  the  other. 

Fig.  2.1(c)  shows  the  electric  field  configuration  when  fast  and  slow 
modes  are  both  present  and  equally  strongly  excited.  We  can  imagine 
the  two  helices  being  excited  by  a  voltage  source  connected  to  the  outer 


(a)   FAST    WAVE    (longitudinal) 


(b)   SLOW    WAVE    (transverse) 


(C)   fast   and    slow    waves    combined    SHOWING    SPATIAL  "BEAT"  PHENOMENON 


Fig.  2.1  — Typical  electric  field  distributions  in  coupled  coaxial  helices  when 
thej^  are  excited  in:  (a)  the  in-phase  or  lonfritudinal  mode,  (b)  the  out-of-phase  or 
transverse  mode,  and  (c)  both  modes  equally. 


COUPLED    HELICES  141 

helix  only  at  the  far  left  side  of  the  sketch.  One,  perfectly  legitimate, 
view  of  the  situation  is  that  the  RF  power,  initially  all  on  the  outer  helix, 
leaks  into  the  inner  helix  because  of  the  coupling  between  them,  and  then 
leaks  back  to  the  outer  helix,  and  so  forth. 

Apart  from  noting  the  appearance  of  the  stationary  spatial  beat  (or 
interference)  phenomenon  these  additional  facts  are  of  interest: 

1)  It  is  a  simple  matter  to  excite  such  a  beat- wave,  for  instance,  by 
connecting  a  lead  to  either  one  or  the  other  of  the  helices,  and 

2)  It  should  be  possible  to  discontinue  either  one  of  the  helices,  at 
points  where  there  is  no  current  (voltage)  on  it,  without  causing  reflec- 
tions. 

2.6  A  Simple  Estimate  of  h  and  x 

How  strong  a  coupling  can  one  expect  from  concentric  helices  in  prac- 
tice? Quantitatively,  this  is  expressed  by  the  values  of  the  coupling  fac- 
tors X  and  h,  which  we  shall  now  proceed  to  estimate. 

A  first  crude  estimate  is  based  on  the  fact  that  slow-wave  fields  are 
known  to  fall  off  in  intensity  somewhat  as  c  where  (3  is  the  phase  con- 
stant of  the  wave  and  r  the  distance  from  the  surface  guiding  the  slow 
wave.  Thus  a  unit  charge  placed,  say,  on  the  inner  helix,  will  induce  a 
charge  of  opposite  sign  and  of  magnitude 

-Pib-a) 

on  the  outer  helix.  Here  h  =  mean  radius  of  the  outer  helix  and  a  = 
mean  radius  of  the  inner.  We  note  that  the  shunt  mutual  admittance 
coupling  factor  is  negative,  irrespective  of  the  directions  in  which  the 
helices  are  wound.  Because  of  the  similarity  of  the  magnetic  and  electric 
field  distributions  a  current  flowing  on  the  inner  helix  will  induce  a  simi- 
larly attenuated  current,  of  amplitude 

on  the  outer  helix.  The  direction  of  the  induced  current  will  depend  on 
whether  the  helices  are  woimd  in  the  same  sense  or  not,  and  it  turns  out 
(as  one  can  verify  by  reference  to  the  low-freciuency  case  of  coaxial 
coupled  coils)  that  the  series  mutual  impedance  coupling  factor  is  nega- 
tive when  the  helices  are  oppositely  wound. 

In  order  to  obtain  the  greatest  possible  coupling  between  concentric 
helices,  both  coupling  factors  should  have  the  same  sign.  This  then  re- 
fiuires  that  the  helices  should  be  wound  in  opposite  directions,  as  has 
been  pointed  out  by  Pierce. 

When  the  distance  between  the  two  helices  goes  to  zero,  that  is  to  say, 


142  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

.if  they  lie  in  the  same  surface,  it  is  clear  that  both  coupling  factors  h  and  x 
will  go  to  unity. 

As  pointed  out  earlier  in  Section  2.3,  the  choice  of  sign  for  h  is  arbi- 
trary. However,  once  a  sign  for  h  has  been  chosen,  the  sign  of  x  is  neces- 
sarily the  opposite  when  the  helices  are  wound  in  the  same  direction,  and 
vice  versa.  We  shall  choose,  therefore, 

the  sign  of  the  latter  depending  on  whether  the  helices  are  wound  in  the 
same  direction  or  not. 

In  the  case  of  unequal  velocities,  (5,  the  propagation  constant,  would 
be  given  by 

1^  =  VM~2  (2.6.2) 

2.7  Strength  of  Coupling  versus  Frequency 

The  exponential  variation  of  coupling  factors  with  respect  to  frequency 
(since  /3  =  co/y)  has  an  important  consequence.  Consider  the  expression 
for  the  coupling  phase  constant 

/3.  =  I3{b  +  x)  (2.3.8) 

or 

l/3e|  =  2/3^"^^'""^  (2.7.1) 

The  coupling  wavelength,  which  is  defined  as 


Ac 
is,  therefore, 


27r 


(2.7.2) 


or 


Xc-  -e 


X,  =  ;^  g(2./x)u.-«)  (2.7.3) 

where  X  is  the  (slowed-down)  RF  wavelength  on  either  helix.  It  is  con- 
venient to  multiply  both  sides  of  (2.7.1)  with  a,  the  inner  helix  radius, 
in  order  to  obtain  a  dimensionless  relation  between  /3c  and  /3: 

^,a  =  2/3ac~^''"''°^""  (2.7.4) 

This  relalion  is  j)l()Ued  on  Fig.  2.2  for  several  values  of  b/a. 


COUPLED   HELICES 


143 


3.00 


2.75 


2.50 


2.25 


2.00 


/3ca 


1.75 


1.50 


1.25 


i.OO 


0.75 


0.50 


0.25 


^^ 

— - 

/ 

/ 
/ 

/ 

/ 

/ 

/ 
/ 
/ 

l-y 

/ 

/ 
/ 

/ 
/ 
J 

/ 

/ 

/ 
/(/Jc3)max 

/ 

/ 

/ 
/ 

/ 

1 

/ 

/ 
/ 
/ 

/ 

^ 

/ 

b 

=  1.5 

/ 

/ 

/ 
/ 

f 

^ 

, 

V 

/ 

^^ 

'" 

\ 

1 

-\ 

"^^^ — 1 

75 

L 

2.0 

■\ 

/ 

"^ 

■-^ 

3.0 

— - 



0.5 


1.5 


2.0 


2.5 

/3a 


3.0 


3.5        4.0 


4.5 


5.0 


Fig.  2.2  —  Coupling  pliase-constant  plotted  as  a  function  of  the  single  helix 
phase-constant  for  synchronous  helices  for  several  values  of  b/a.  These  curves 
are  based  on  simple  estimates  made  in  Section  2.7. 


There  are  two  opposing  tendencies  determining  the  actual  physical 
length  of  a  coupling  beat-wavelength: 

1)  It  tends  to  grow  with  the  RF  wavelength,  being  proportional  to  it 
in  the  first  instance; 

2)  Because  of  the  tighter  coupling  possible  as  the  RF  wavelength 
increases  in  relation  to  the  heli.x-to-helix  distance,  the  coupling  beat- 
wavelength  tends  to  shrink. 

Therefore,  there  is  a  region  where  these  tendencies  cancel  each  other, 
and  where  one  would  expect  to  find  little  change  of  the  coupling  beat- 
wavelength  for  a  considerable  change  of  RF  freciuency.  In  other  words, 
the  "bandwidth"  over  which  the  beat-wavelength  stays  nearly  constant 
can  be  large. 

This  is  a  situation  naturally  very  desirable  and  favorable  for  any 
device  in  which  we  rely  on  power  transfer  from  one  helix  to  the  other  by 


144  THE    BELL    SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

means  of  a  length  of  overlap  between  them  an  integral  number  of  half 
beat-wavelengths  long.  Ob^'iously,  one  will  design  the  helices  in  such  a 
way  as  to  take  advantage  of  this  situation. 

Optimiun  conditions  are  easily  obtained  by  dijfferentiating  ^c  with 
respect  to  (3  and  setting  d^c/d^  equal  to  zero.  This  gives  for  the  optimum 
conditions 


^opt    — 


1 


b  —  a 


(2.7.5) 


or 


Pc  opt 


2e 


h  —  a 


=  2e  ')8opt 


(2.7.6) 


Equation  (2.7.5),  then,  determines  the  ratio  of  the  helix  radii  if  it  is  re- 
quired that  deviations  from  a  chosen  operating  frequency  shall  have 
least  effect. 

2.8  Field  Solutions 

In  treating  the  problem  of  coaxial  coupled  helices  from  the  transmis- 
sion line  point  of  view  one  important  fact  has  not  been  considered, 
namely,  the  dispersive  character  of  the  phase  constants  of  the  separate 
helices,  /3i  and  fS-i  .  By  dispersion  we  mean  change  of  phase  velocity  with 
frequency.  If  the  dispersion  of  the  inner  and  outer  helices  were  the  same 
it  would  be  of  little  consequence.  It  is  well  known,  however,  that  the 
dispersion  of  a  helical  transmission  line  is  a  function  of  the  ratio  of  helix 
radius  to  wavelength,  and  thus  becomes  a  parameter  to  be  considered. 
When  the  theory  of  wave  propagation  on  a  helix  was  solved  by  means  of 
Maxwell's  equations  subject  to  the  boundary  condition  of  a  helically 
conducting  cylindrical  sheath,  the  phenomenon  of  dispersion  first  made 
its  appearance.  It  is  clear,  therefore,  that  a  more  complete  theory  of 


/i 


'V^       'TV 


Fig.  2.3  —  ShoMtli  liolix  arrangement  on  which  the  field  equations  are  based. 


COUPLED    HELICES  145 

coupled  helices  will  require  similar  treatment,  namely,  Maxwell's  equa- 
tions solved  now  with  the  boundary  conditions  of  two  cylindrical  heli- 
cally conducting  sheaths.  As  shown  on  Fig.  2.3,  the  inner  helix  is  specified 
by  its  radius  a  and  the  angle  1^1  made  by  the  direction  of  conductivity 
with  a  plane  perpendicular  to  the  axis;  and  the  outer  helix  by  its  radius 
h  (not  to  be  confused  with  the  mutual  coupling  coefficient  5)  and  its 
corresponding  pitch  angle  i/'-j  .  We  note  here  that  oppositely  wound  helices 
require  opposite  signs  for  the  angles  \f/i  and  i/'o  ;  and,  further,  that  helices 
with  equal  phase  velocities  will  ha\'e  pitch  angles  of  about  the  same 
absolute  magnitude. 

The  method  of  solving  Maxwell's  equations  subject  to  the  above  men- 
tioned boundary  conditions  is  given  in  Appendix  I.  We  restrict  our- 
selves here  to  giving  some  of  the  results  in  graphical  form. 

The  most  universally  used  parameter  in  traveling-wave  tube  design  is 
a  combination  of  parameters: 

/3oa  cot  \pi 

where  (So  =  27r/Xo ,  Xo  being  the  free-space  wavelength,  a  the  radius  of 
the  inner  helix,  and  xpi  the  pitch  angle  of  the  inner  helix.  The  inner  helix 
is  chosen  here  in  preference  to  the  outer  helix  because,  in  practice,  it  will 
be  part  of  a  traveling-wave  tube,  that  is  to  say,  inside  the  tube  envelope. 
Thus,  it  is  not  only  less  accessible  and  changeable,  but  determines  the 
important  aspects  of  a  traveling-wave  tube,  such  as  gain,  power  output, 
and  efficiency. 

The  theory  gives  solutions  in  terms  of  radial  propagation  constants 
which  we  shall  denote  jt  and  yt  (bj^  analogy  with  the  transverse  and 
longitudinal  modes  of  the  transmission  line  theory).  These  propagation 
constants  are  related  to  the  axial  propagation  constants  ^t  and  j3(  by 

Of  course,  in  transmission  line  theory  there  is  no  such  thing  as  a  radial 
propagation  constant.  The  propagation  constant  derived  there  and  de- 
noted r  corresponds  here  to  the  axial  propagation  constant  j^.  By 
analogy  with  (2.4.5)  the  beat  phase  constant  should  be  written 

How^ever,  in  practice  ^0  is  usually  much  smaller  than  j3  and  Ave  can  there- 
fore write  with  little  error 

iSfc  =  7e  —  li 
for  the  beat  phase  constant.  For  practical  purposes  it  is  convenient  to 


146  THE    BELL    SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


J.OU 

_^^ 



1 

3i.Z0 

COT  ^2    _        „„„ 

;:^ 

^ 

COTV'i 

-0.90.. 

\ 

!^ 

-0.82,^ 

^ 

:J^ 

2.80 

>^ 

<r 

Q  2.40 
^2.00 

^ 

/. 

^ 

// 

/ 

^ 

|=,.25 

1.  60 

1.20 

i 

/// 

0.80 

0.5 


1.0 


2.0  2.5 

/io  a  COT  ^, 


3.0 


3.5 


4.0 


4.5 


Fig.  2.4.1  —  Beat  phase-constant  plotted  as  a  function  of  /3oa  cot  i^i  •  These 
curves  result  from  the  solution  of  the  field  equations  given  in  the  appendix.  For 
hi  a  =  1.25. 

normalize  in  terms  of  the  inner  helix  radius,  a: 


jSbO 


7<a  —  7/a 


This  has  been  plotted  as  a  function  of  /5o  a  cot  i/'i  in  Fig.  2.4,  which 
should  be  compared  with  Fig.  2.2.  It  will  be  seen  that  there  is  considerable 
agreement  between  the  results  of  the  two  methods, 

2.9  Bifilar  Helix 

The  failure  of  the  transmission  line  theory  to  take  into  account  dis- 
persion is  well  illustrated  in  the  case  of  the  bifilar  helix.  Here  we  have 
two  identical  helices  wound  in  the  same  sense,  and  at  the  same  radius. 
If  the  two  wires  are  fed  in  phase  we  have  the  normal  mode  characterized 
by  the  sheath  helix  model  whose  propagation  constant  is  the  familiar 
Curve  A  of  Fig.  2.5.  If  the  two  wires  of  the  helix  are  fed  out  of  phase  we 
have  the  bifilar  mode;  and,  since  that  is  a  two  wure  transmission  system, 
we  shall  have  a  TEM  mode  which,  in  the  absence  of  dielectric,  propa- 
gates along  the  wire  with  the  velocity  of  light.  Hence,  the  propagation 
constant  for  this  mode  is  simplj'  /3oa  cot  \p  and  gives  rise  to  the  horizontal 


COUPLED    HELICES 


147 


1.80 


1.60 


(0 

n  1.40 
<5. 


I 
to 

t.OO 


0.80 


0.60 


b. 

^ 

^>. 

"^ 

"a"'" 

A 

s 

^ 

N. 

\. 

\^ 

i 

& 

\ 

\ 

^ 

■^ 

0.82 

w 

^   =  -0.98 

COT^, 

^ 

0.90 

^V 

^ 

// 

/ 

\ 

\ 

v 

J, 

/ 

\ 

\ 

\, 

t 

\ 

f 

\ 

f 

0.5 


1.0 


1.5 


2.0  2.5 

/3oaCOTi^, 


3.0 


3.5 


4.0 


4.5 


Fig.  2.4.2  —  Beat  phase-constant  plotted  as  a  function  of  /3oa  cot  ^i-i  .  These 
curves  result  from  the  solution  of  the  field  equations  given  in  the  appendix.  For 
hia  =  1.5. 

line  of  Curve  B  in  Fig.  2.5.  Again  the  coupling  phase  constant  j3c  is  given 
by  the  difference  of  the  individual  phase  constants: 


^cO-  —  /3oa  cot  \f/  —  ya 


(2.9.1) 


which  is  plotted  in  Fig.  2.6.  Now  note  that  when  /So  <3C  7  this  equation  is 
accurate,  for  it  represents  a  solution  of  the  field  equations  for  the  helix. 

From  the  simple  unsophisticated  transmission  line  point  of  view  no 
coupling  between  the  two  helices  would,  of  course,  have  been  expected, 
since  the  two  helices  are  identical  in  every  way  and  their  mutual  capacity 
and  inductance  should  then  be  equal  and  opposite. 

Experiments  confirm  the  essential  correctness  of  (2.9.1).  In  one  experi- 
ment, which  was  performed  to  measure  the  coupling  wavelength  for  the 
l)ifilar  helices,  we  used  helices  with  a  cot  1/'  =  3.49  and  a  radius  of  0.036 
cm  which  gave  a  value,  at  3,000  mc,  of  ^oa  cot  i^  =  0.51 .  In  these  experi- 
ments the  coupling  length,  L,  defined  by 

(/3oa  cot  xp  —  7a)  —  =   TT 
a 

was  measured  to  be  15.7o  as  compared  to  a  value  of  13.5a  from  Fig.  2.6. 
At  4,000  mc  the  measured  coupling  length  was  14.6a  as  compared  to 


148  THE    BELL   vSYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


1.20 

b 
a 

1.76 

^ 

^^ 

^ 

^ 

X, 

/ 

y 

^ 

^ 
S. 

X 

1.00 

/ 

V 

\ 

^. 

-\ 

/ 

/ 

\ 

P> 

•^.82 

0.80 

r — 

\ 

^ 

<5. 

COT^ 
COT^ 

N 

^  =  -0.9 
1 

k^ 

"^^^ 

0.90 

(0  0.60 

a  >s^ 

X 

<0 

'^  0.40 

^ 

. 

0.20 

0 
< 

D 

0 

5 

1 

0 

1 

5 

2 

.0 

2 

.5 
^1 

3.0 

3.5 

4.0 

4 

Fig.  2.4.3  —  Beat  phase-constant  plotted  as  a  function  of  ^^a  cot  -^x  .  These 
curves  result  from  the  solution  of  the  field  equations  given  in  the  appendix.  For 
hi  a  =  1.75. 

12.6a  computed  from  Fig.  2.6,  thus  confirming  the  theoretical  prediction 
rather  well.  The  slight  increase  in  coupling  length  is  attributable  to  the 
dielectric  loading  of  the  helices  which  were  supported  in  quartz  tubing. 
The  dielectric  tends  to  decrease  the  dispersion  and  hence  reduce  /3,.  .  This 
is  discussed  further  in  the  next  section. 


2.10  Effect  of  Dielectric  Material  hetween  Helices 

In  many  cases  which  are  of  interest  in  practice  there  is  dielectric  ma- 
terial between  the  helices.  In  particular  when  coupled  helices  are  used 
with  traveling-wave  tubes,  the  tube  envelope,  which  may  be  of  glass, 
quartz,  or  ceramic,  all  but  fills  the  space  between  the  two  helices. 

It  is  therefore  of  interest  to  know  whether  such  dielectric  makes  any 
difference  to  the  estimates  at  which  we  arrived  earlier.  We  should  not  be 
surprised  to  find  the  coupling  strengthened  by  the  presence  of  the  di- 
electric, because  it  is  known  that  dielectrics  tend  to  rob  RF  fields  from 
the  surrounding  space,  leading  to  an  increase  in  the  energy  flow  through 
the  dielectric.  On  the  other  hand,  tlio  dielectric  tends  to  bind  the  fields 
closer  to  the  conducting  medium.  To  find  a  qualitative  answer  to  this 
question  we  have  calculated  the  relative  coupling  phase  constants  for 
two  sheath  helices  of  infinite  radius  separated  by  a  distance  "d"  for  1) 


COUPLED    HELICES 


149 


1.00 

b 

-a-^.u 

^ 

^^ 

^ 

^ 

j^ 

^ 

COT  Tp2 

^ 

^ 

C  0.60 

)^ 

1    0.40 
m 

y 

^ 

COT  }^, 

^ 1 

> 

-^ 

S^ 

^ 

. , 

— 

-0. 

90 

=- 

-- 

V 

^, 

i 

^ 



^0^ 

98 

0.20 

1 

0 

1 

( 

3 

0 

5 

1 

0 

1 

5 

2 

0 

>oac 

2.5 

3 

.0 

3 

.5 

4 

0 

4. 

Fig.  2.4.4  —  Beat  phase-constant  plotted  as  a  function  of  /3oa  cot  ^i  .  These 
curves  result  from  the  solution  of  the  field  equations  given  in  the  appendix.  For 
b/a  =  2.0. 

the  case  with  dielectric  between  them  having  a  relative  dielectric  con- 
stant e'  =  4,  and  2)  the  case  of  no  dielectric.  The  pitch  angles  of  the  two 
helices  were  \p  and  —xp,  respectively;  i.e.,  the  helices  were  assumed  to  be 
synchronous,  and  wound  in  the  opposite  sense. 
■  Fig.  2.7  shows  a  plot  of  the  ratio  of  /3,,.//3,  to  ^d^  versus  /3o  (f//2)  cotiA, 


1.00 


0.80 


to 
n 

«5.  0.60 

II 
m 

i    0.40 


0.20 


b 

a-o.u 

^y 

^ 

>< 

^ 

y 

COT  ^2 

^ 

\>^ 

^-- 

COT  5^, 

-^ 

r 

/, 

^ 
^ 

==^ 

"^^ 

N^ 

^^ 

y^ 

^ 

^ 

^ 

f/ 

^ 

^ 

N. 

^ 

-c 

).90 

-^ 

r 

\ 

-^ 

_ 

-o.s 

?8 

, 

^ 

0.5 


1.0 


f.5 


2.0  2.5 

/JoacoT;^, 


3.0 


3.5 


4.0 


4.5 


Fig.  2.4.5  — ■  Beat  phase-constant  plotted  as  a  function  of  (3o«  cot  ^\  .  These 
curves  result  from  the  solution  of  the  field  equations  given  in  the  appendix.  For 
Va  =  3.0. 


150  THE    BELL    SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


2.4 


0.5       ).0       1.5      2.0      2. 


5      3.0      3.5 
/3oaCOT^ 


4.0      4.5 


Fig.  2.5  —  Propagation  constants  for  a  bifilar  helix  plotted  as  a  function  of 
/3oa  cot  i/-!  .  The  curves  illustrate,  (A)  the  dispersive  character  of  the  in-phase 
mode  and,  (B)  the  non-dispersive  character  of  the  out-of -phase  mode. 

where  ^^  is  the  coupling  phase-constant  in  the  presence  of  dielectric, 
/3j  is  the  phase-constant  of  each  helix  alone  in  the  presence  of  the  same 
dielectric,  ^c  is  the  coupling  phase-constant  with  no  dielectric,  and  (3  is 
the  phase  constant  of  each  helix  in  free  space.  In  many  cases  of  interest 
/3o(d/2)  cot  lA  is  greater  than  1.2.  Then 


3£    +  1" 
_2£'  +  2_ 


g—(v'2« '+2-2)^0  (dl2)  cot  \l/ 


(2.10.1) 


Appearing  in  the  same  figure  is  a  similar  plot  for  the  case  when  there  is  a 
conducting  shield  inside  the  inner  helix  and  outside  the  outer,  and 
separated  a  distance,  "s,"  from  the  helices.  Note  that 

c?  =  6  —  a. 


It  appears  from  these  calculations  that  the  effect  of  the  presence  of 
dielectric  between  the  helices  depends  largely  on  the  parameter  /So  (d/2) 
cot  \{/.  For  values  of  this  parameter  larger  than  0.3  the  coupling  wave- 
length tends  to  increase  in  terms  of  circuit  wavelength.  For  values  smaller 
than  0.3  the  opposite  tends  to  happen.  Note  that  the  curve  representing 
(2.10.1)  is  a  fair  approximation  down  to /3o(c?/2)  cot  i/'  =  0.6  to  the  curve 
representing  the  exact  solution  of  the  field  equations.  J.  W.  Sullivan,  in 
unpublished  work,  has  drawn  similar  conclusions. 


COUPLED   HELICES 


151 


2.11  The  Conditions  for  Maximum  Power  Transfer 

The  transmission  line  theory  has  led  us  to  expect  that  the  most  efficient 
power  transfer  will  take  place  if  the  phase  velocities  on  the  two  helices, 
prior  to  coupling,  are  the  same.  Again,  this  would  be  true  were  it  not  for 
the  dispersion  of  the  helices.  To  evaluate  this  effect  we  have  used  the 
field  equation  to  determine  the  parameter  of  the  coupled  helices  which 
gives  maximum  power  transfer.  To  do  this  we  searched  for  combinations 
of  parameters  which  give  an  equal  current  flow  in  the  helix  sheath  for 
either  the  longitudinal  mode  or  the  transverse  mode.  This  was  suggested 
by  L.  Stark,  who  reasoned  that  if  the  currents  were  equal  for  the  indi- 
vidual modes  the  beat  phenomenon  would  give  points  of  zero  RF  current 
on  the  helix. 

The  values  of  cot  T/'2/cot  4/i  which  are  required  to  produce  this  condi- 
tion are  plotted  in  Fig.  2.8  for  various  values  of  b/a.  Also  there  are  shown 
values  of  cot  ^2/cot  \{/i  required  to  give  equal  axial  velocities  for  the  helices 
before  they  are  coupled.  It  can  be  seen  that  the  uncoupled  velocity  of  the 
inner  helix  must  be  slightly  slower  than  that  of  the  outer. 

A  word  of  caution  is*  necessary  for  these  curves  have  been  plotted 
without  considering  the  effects  of  dielectric  loading,  and  this  can  have  a 
rather  marked  effect  on  the  parameters  which  we  have  been  discussing. 
The  significant  point  brought  out  by  this  calculation  is  that  the  optimum 


u.^o 

r 

N 

0.24 
0.20 

/ 

\ 

\ 

/ 

N 

<D 

/ 

\, 

/ 

N 

/ 

N^ 

0 

^  0.12 

\ 

\^ 

~j 

■^v 

CD 

^■^^^ 

0 

0.08 

f- 

^- 

-^ 

0.04 

0 

04         0.8  1.2  1.6  2.0  2.4 

/3oaCOT  J^, 


2.8 


3.2 


3.6 


4.0 


Fig.  2.6  —  The  coupling  phase-constant  which  results  from  the  two  possible 
modes  of  propagation  on  a  bifilar  helix  shown  as  a  function  of  jSoo  cot  i/-!  . 


152  THE    BELL   SYSTEM   TECHXICAL   JOURNAL,    JANUARY    1956 

2.6 

2.4 

2.2 
2.0 

i.8 

1.6 


u 


1.4 


i.2 


1.0 


0.8 


0.6 


0.4 


0.2 


PROPAGATION 

DIRECTION 

\ 

\ 

^, 

\ 

L 

VA 

\ 

s 

PLANE  SHEATH  -^^^^"'^  XdiELECTRIC, 
HELICES                         \^^r                e' 
CONDUCTING 
SHIELD 

\ 

\ 

\ 

s=oo 

\ 

\, 

APPROXIMATION 

^^ 

^, 

s 

\ 

"N 

'^ 

\ 

^ 

"^^^ 

■^^ 

o.t 


0.2        0.3 


0.4        0.5 


0.6 


0.7 


0.8 


0.9 


1.0 


1.1 


1.2 


/iofcOT^^ 


Fig.  2.7 — ^  The  effect  of  dielectric  material  between  coupled  infinite  radius 
sheath  helices  on  their  relative  coupling  phase-constant  shown  as  a  function  of 
fiod/2  cot  \pi  .  The  effect  of  shielding  on  this  relation  is  also  indicated. 

condition  for  coupling  is  not  necessarily  associated  with  equal  \'elocities 
on  the  uncoupled  helices. 


2.12  Mode  Impedance 

Before  leaving  the  general  theor_y  of  coupled  helices  something  should 
be  said  regarding  the  impedance  their  modes  present  to  an  electron  beam 
traveling  either  along  their  axis  or  through  the  annular  space  between 
them.  The  field  solutions  for  cross  woimd,  coaxiall}^  coupled  helices, 
which  are  given  in  Appendix  I,  have  been  used  to  compute  the  imped- 
ances of  the  transverse  and  longitudinal  modes.  The  impedance,  /v,  is 
defined,  as  usual,  in  terms  of  the  longitudinal  field  on  the  axis  and  the 
power  flow  along  the  system. 


COUPLED    HELICES 


153 


K  = 


F{ya) 


In  Fig.  2.9,  Fiya),  for  various  I'atios  of  inner  to  outer  radius,  is  plotted 
for  both  the  transverse  and  longitudinal  modes  together  with  the  value 
of  F{ya)  for  the  single  helix  {b/a  =  co).  We  see  that  the  longitudinal 
mode  has  a  higher  impedance  with  cross  wound  coupled  helices  than 
does  a  single  helix.  We  call  attention  here  to  the  fact  that  this  is  the 
same  phenomenon  which  is  encountered  in  the  contrawound  helix^,  where 
the  structure  consists  of  two  oppositely  wound  helices  of  the  same  radius. 
As  defined  here,  the  transverse  mode  has  a  lower  impedance  than  the 
single  helix.  This,  however,  is  not  the  most  significant  comparison;  for 
it  is  the  transverse  field  midway  between  helices  which  is  of  interest  in 
the  transverse  mode.  The  factor  relating  the  impedance  in  terms  of  the 
transverse  field  between  helices  to  the  longitudinal  field  cni  the  axis  is 
Er  (f)/Ei(0),  where  f  is  the  radius  at  which  the  longitudinal  component 
of  the  electric  field  E^ ,  is  zero  for  the  transverse  mode.  This  factor, 
plotted  in  Fig.  2.10  as  a  function  of  /3oa  cot  \l/r ,  shows  that  the  impedance 
in.  terms  of  the  transverse  field  at  f  is  interestingly  high. 


1.00 


0.72 


1.6  2.0  2.4 

/3o  a  COT   Ifi 


4.0 


Fig.  2.8  —  The  values  of  cot  ^^./cot  \pi  required  for  complete  power  transfer 
plotted  as  a  function  of  /3tia  cot  \pi  for  several  values  of  b/a.  For  comparison,  the 
value  of  cot  ^2/cot  \//i  required  for  equal  velocities  on  inner  and  outer  helices  is  also 
shown. 


154  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


F(ra) 


7.0 
6.5 
6.0 
5.5 
5.0 
4.5 
4.0 

3.5 

• 
3.0 

2.5 

2.0 

1.5 

1.0 

0.5 


0.5  (.0         (.5        2.0        2.5        3.0        3.5        4.0        4.5        5.0 

/SoaCOTii', 

Fig.  2.9  — ■  Impedance  parameter,  F(ya),  associated  with  both  transverse  and 
longitudinal  modes  shown  for  several  values  of  b/a.  Also  shown  is  F{ya)  for  a 
single  helix. 

It  is  also  of  interest  to  consider  the  impedance  of  the  longitudinal 
mode  in  terms  of  the  longitudinal  field  between  the  two  helices.  The 
factor,  ^/(f)/£'/(0),  relating  this  to  the  axial  impedance  is  plotted  in 
Fig.  2.11.  We  see  that  rather  high  impedances  can  also  be  obtained  with 
the  longitudinal  field  midway  between  helices.  This,  in  conjunction  with 
a  hollow  electron  beam,  should  provide  efficient  amplification. 


LONGITUDINAL  WAVE 

V 

COT  U/2 

\ 

\^.    V 

\ 

\ 

\        "^ 

^=-0.90 

k      \ 

\ 

COT  U/^ 

V       \ 

\ 

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b.ooV        ^ 

\ 

\ 
\ 

a 

\ 

\ 

^ 

\ 

\ 

\ 

\-o\ 

\ 
\ 

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L                 \ 

i 

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yp 

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XT' 

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\  ' 

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k 

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1=1.2^ 

^^^ 

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^^ 

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>., 

'- 

^^ 

^ 

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^^ 

^==^ 

3.    APPLICATION    OF    COUPLED   HELICES 


When  we  come  to  describe  devices  which  make  use  of  coupled  helices 
we  find  that  they  fall,  quite  naturally,  into  two  separate  classes.  One 


COUPLED   HELICES 


155 


class  contains  those  devices  which  depend  on  the  presence  of  only  one  of 
the  two  normal  modes  of  propagation.  The  other  class  of  devices  depends 
on  the  simultaneous  presence,  in  roughly  equal  amounts,  of  both  normal 
modes  of  propagation,  and  is,  in  general,  characterized  by  the  words 
"spatial  beating."  Since  spatial  beating  implies  energy  surging  to  and 
fro  between  inner  and  outer  helix,  there  is  no  special  problem  in  exciting 
both  modes  simultaneously.  Power  fed  exclusively  to  one  or  the  other 


/bo  a  COT  jfi, 


Fiji;.  2.10  —  The  relation  l)et\veen  the  impedance  in  terms  of  the  transverse 
field  between  conpled  helices  excited  in  the  out-of -phase  mode,  and  the  impedance 
in  terms  of  the  longitudinal  field  on  the  axis  shown  as  a  function  of  /3oa  cot  tpi  . 


156 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 


helix  will  inevitably  excite  both  modes  equally.  When  it  is  desired  to 
excite  one  mode  exclusively  a  more  difficult  problem  has  to  be  solved. 
Therefore,  in  section  3.1  we  shall  first  discuss  methods  of  exciting  one 
mode  only  before  going  on  to  discuss  in  sections  3.2  and  3.3  devices 
using  one  mode  only. 

In  section  3.4  we  shall  discuss  devices  depending  on  the  simultaneous 
presence  of  both  modes. 

3.1  Excitation  of  Pure  Modes 

3.1.1  Direct  Excitation 

In  order  to  set  up  one  or  the  other  normal  mode  on  coupled  helices, 
voltages  with  specific  phase  and  amplitudes  (or  corresponding  currents) 


E|(f) 
E|(o) 


10^ 
5 

10^ 


10^ 


10' 


10 


10" 


COT 

ip? 

■ —  =  -0.90 

COT  i^, 

1 

/ 

/ 

1 

' 

l-.o/ 

1 

L 

L 

1 

/ 

J 

l\.2b 

/ 

J 

/    J 

/ 

^ 

'^ 

3  A 

/ho  a  COT  1fi^ 


Fig.  2.11  — -The  relation  Ijetween  the  impedance  in  terms  of  the  longitudinal 
field  between  couj)led  helices  excited  in  the  in-phase  mode,  and  the  impedance  in 
terms  of  the  longitudinal  field  on  the  axis  shown  as  a  function  of  /3offl  cot  \pi  . 


COUPLED   HELICES  157 

have  to  be  supplied  to  each  helix  at  the  input  end.  A  natural  way  of  doing 
this  might  be  by  means  of  a  two-conductor  balanced  transmission  line 
(Lecher-line),  one  conductor  being  connected  to  the  inner  helix,  the  other 
to  the  outer  helix.  Such  an  arrangement  would  cause  something  like  the 
transverse  (-| — )  mode  to  be  set  up  on  the  helices.  If  the  two  con- 
ductors and  the  balanced  line  can  be  shielded  from  each  other  starting 
some  distance  from  the  helices  then  it  is,  in  principle,  possible  to  intro- 
duce arbitrary  amounts  of  extra  delay  into  one  of  the  conductors.  A  delay 
of  one  half  period  would  then  cause  the  longitudinal  (  +  +  )  mode  to  be 
set  up  in  the  helices.  Clearly  such  a  coupling  scheme  would  not  be 
broad-band  since  a  frequency-independent  delay  of  one  half  period  is  not 
realizable. 

Other  objections  to  both  of  these  schemes  are:  Balanced  lines  are  not 
generally  used  at  microwave  frequencies;  it  is  difficult  to  bring  leads 
through  the  envelope  of  a  TWT  without  causing  reflection  of  RF  energy 
and  without  unduly  encumbering  the  mechanical  design  of  the  tube  plus 
circuits;  both  schemes  are  necessarily  inexact  because  helices  having 
different  radii  will,  in  general,  require  different  voltages  at  either  input 
in  order  to  be  excited  in  a  pure  mode. 

Thus  the  practicability,  and  success,  of  any  general  scheme  based  on 
the  existence  of  a  pure  transverse  or  a  pure  longitudinal  mode  on  coupled 
helices  will  depend  to  a  large  extent  on  whether  elegant  coupling  means 
are  available.  Such  means  are  indeed  in  existence  as  will  be  shown  in  the 
next  sections. 

3.1.2  Tapered  Coupler 

A  less  direct  but  more  elegant  means  of  coupling  an  external  circuit 
to  either  normal  mode  of  a  double  helix  arrangement  is  by  the  use  of  the 
so-called  "tapered"  coupler.^'  ^'  ^^  By  appropriately  tapering  the  relative 
propagation  velocities  of  the  inner  and  outer  helices,  outside  the  inter- 
action region,  one  can  excite  either  normal  mode  by  coupling  to  one 
helix  only. 

The  principle  of  this  coupler  is  based  on  the  fact  that  any  two  coupled 
transmission  lines  support  two,  and  only  two,  normal  modes,  regardless 
of  their  relative  phase  velocities.  These  normal  modes  are  characterized 
by  unequal  wave  amplitudes  on  the  two  lines  if  the  phase  velocities  are 
not  equal.  Indeed  the  greater  the  phase  velocity  difference  and /or 
the  smaller  the  coupling  coefficient  between  the  lines,  the  more  their 
wave  amplitudes  diverge.  Furthermore,  the  wave  amplitude  on  the  line 
with  the  slower  phase  velocity  is  greater  for  the  out-of-phase  or  trans- 
verse normal  mode,  and  the  wave  amplitude  on  the  faster  line  is  greater 


158  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    195G 

for  the  longitudinal  normal  mode.  As  the  ratio  of  phase  constant  to 
coupling  constant  approaches  infinity,  the  ratio  of  the  wave  amplitudes 
on  the  two  lines  does  also.  Finally,  if  the  phase  velocities  of,  or  coupling 
between,  two  coupled  helices  are  changed  gradually  along  their  length 
the  normal  modes  existing  on  the  pair  roughly  maintain  their  identity 
evan  though  they  change  their  character.  Thus,  by  properly  tapering  the 
phase  velocities  and  coupling  strength  of  any  two  coupled  helices  one 
can  cause  the  two  normal  modes  to  become  two  separate  waves,  one 
existing  on  each  helix. 

For  instance,  if  one  desires  to  extract  a  signal  propagating  in  the  in- 
phase,  or  longitudinal,  normal  mode  from  two  concentric  helices  of  equal 
phase  velocity,  one  might  gradually  increase  the  pitch  of  the  outer  helix 
and  decrease  that  of  the  inner,  and  at  the  same  time  increase  the  diameter 
of  the  outer  helix  to  decrease  the  coupling,  until  the  longitudinal  mode 
exists  as  a  wave  on  the  outer  helix  only.  At  such  a  point  the  outer  helix 
may  be  connected  to  a  coaxial  line  and  the  signal  brought  out. 

This  kind  of  coupler  has  the  advantage  of  being  frequency  insensitive ; 
and,  perhaps,  operable  over  bandwidths  upwards  of  two  octaves.  It 
has  the  disadvantage  of  being  electrically,  and  sometimes  physically, 
quite  long. 

3.1.3  Stepped  Coupler 

There  is  yet  a  third  way  to  excite  only  one  normal  mode  on  a  double 
helix.  This  scheme  consists  of  a  short  length  at  each  end  of  the  outer  helix, 
for  instance,  which  has  a  pitch  slightly  different  from  the  rest.  This 
has  been  called  a  "stepped"  coupler. 

The  principle  of  the  stepped  coupler  is  this:  If  two  coupled  transmis- 
sion lines  have  unlike  phase  velocities  then  a  wave  initiated  in  one  line 
can  never  be  completely  transferred  to  the  other,  as  has  been  shown  in 
Section  2.4.  The  greater  the  velocity  difference  the  less  will  be  the  maxi- 
mum transfer.  One  can  choose  a  velocity  difference  such  that  the  maxi- 
mum power  transfer  is  just  one  half  the  initial  power.  It  is  a  characteristic 
of  incomplete  power  transfer  that  at  the  point  where  the  maximum  trans- 
fer occurs  the  waves  on  the  two  lines  are  exactly  either  in-phase  or  out-of- 
phase,  depending  on  which  helix  was  initially  excited.  Thus,  the  condi- 
tions for  a  normal  mode  on  two  equal-velocity  helices  can  be  produced 
at  the  maximum  transfer  point  of  two  unlike  velocity  helices  by  initiating 
a  wave  on  only  one  of  them.  If  at  that  point  the  helix  pitches  are  changed 
to  give  equal  phase  velocities  in  both  helices,  with  equal  current  or  volt- 
age amplitude  on  both  helices,  either  one  or  the  other  of  the  two  normal 
modes  will  be  propagated  on  the  two  helices  from  there  on.  Although  the 


COUPLED   HELICES  159 

pitch  and  length  of  such  a  stepped  coupler  are  rather  critical,  the  re- 
quirements are  indicated  in  the  equations  in  Section  2.4. 

The  useful  bandwidth  of  the  stepped  coupler  is  not  as  great  as  that 
of  the  tapered  variety,  but  may  be  as  much  as  an  octave.  It  has  however 
the  advantage  of  being  very  much  shorter  and  simpler  than  the  tapered 
coupler. 

3.2  Low-Noise  Transverse-Field  Amplifier 

r  One  application  of  coupled  helices  which  has  been  suggested  from  the 
very  beginning  is  for  a  transverse  field  amplifier  with  low  noise  factor. 
In  such  an  amplifier  the  EF  structure  is  required  to  produce  a  field  which 
is  purely  transverse  at  the  position  of  the  beam.  For  the  transverse  mode 
there  is  always  such  a  cylindrical  surface  where  the  longitudinal  field  is 
zero  and  this  can  be  obtained  from  the  field  equation  of  Appendix  II. 
In  Fig.  3.1  we  have  plotted  the  value  of  the  radius  f  at  which  the  longi- 
tudinal field  is  zero  for  various  parameters.  The  significant  feature  of 
this  plot  is  that  the  radius  which  specifies  zero  longitudinal  field  is  not 
constant  with  frequency.  At  frequencies  away  from  the  design  frequency 
the  electron  beam  will  be  in  a  position  where  interaction  with  longitudinal 
components  might  become  important  and  thus  shotnoise  power  will  be 
introduced  into  the  circuit.  Thus  the  bandwidth  of  the  amplifier  over 
which  it  has  a  good  noise  factor  would  tend  to  be  limited.  However,  this 
effect  can  be  reduced  by  using  the  smallest  practicable  value  of  b/a. 

Section  2.12  indicates  that  the  impedance  of  the  transverse  mode  is 
very  high,  and  thus  this  structure  should  be  well  suited  for  transverse 
field  amplifiers. 

3.3  Dispersive  Traveling-Wave  Tube 

Large  bandwidth  is  not  always  essential  in  microwave  amplifiers.  In 
particular,  the  enormous  bandwidth  over  which  the  traveling-wave  tube 
is  potentially  capable  of  amplifying  has  so  far  found  little  application, 
while  relatively  narrow  bandwidths  (although  quite  wide  by  previous 
standards)  are  of  immediate  interest.  Such  a  relatively  narrow  band,  if 
it  is  an  inherent  electronic  property  of  the  tube,  makes  matching  the 
tube  to  the  external  circuits  easier.  It  may  permit,  for  instance,  the  use 
of  non-reciprocal  attenuation  by  means  of  ferrites  in  the  ferromagnetic 
resonance  region.  It  obviates  filters  designed  to  deliberately  reduce  the 
band  in  certain  applications.  Last,  but  not  least,  it  offers  the  possibility 
of  trading  bandwidth  for  gain  and  efficiency. 

A  very  simple  method  of  making  a  traveling-wave  tube  narrow-band 


160  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


0.5 


1.W 

1.8 

^ 

^ 

1.7 

COT  \p. 

_^ 

^ 

^ 

<^ 

T^  =  -0. 

COT  ^, 

82^ 

^ 

^ 

^ 

^ 

l=- 

1.6 

^ 

^ 

^ 

-0.90 

^ 

^ 

^ 

^ 

* 

1.5 

— - 



-0.9 

8 

' 

^-'     ' 

^ 

^ 

1.4 

-^ 

COT  ^i'2 

^-^  =  -0.82 

COT  UJ^                 , 

-0.9j 

, 





^- 

1.3 



. 

— 

"ZH 

'          1 
-0.98 

1.2 

_ 

"71 

|  =  ,.25 

— 

^= 

CO' 

r  1//. 

— T-" 

H 

COT  5^, 

=  -0.82  - 
-0.90  ■ 

'                  / 
^ 

1 
/ 

/ 

i.n 

-0 

.98 

~ 

1.0 


1.5 


2.0 


2.5  3.0 

/3o  a  COT  j^. 


3.5 


4.0 


4.5 


5.0 


Fig.  3.1  —  The  radius  r  at  which  the  longitudinal  field  is  zero  for  transversely 
excited  coupled  coaxial  helices. 


is  by  using  a  dispersive  circuit,  (i.e.  one  in  which  the  phase  velocity  varies 
significantly  with  frequency).  Thus,  we  obtain  an  amplifier  that  can  be 
limed  by  varying  the  beam  voltage;  being  dispersive  we  should  also 
expect  a  low  group  velocity  and  therefore  higher  circuit  impedance. 

Calculations  of  the  phase  velocities  of  the  normal  modes  of  coupled 
concentric  helices  presented  in  the  appendix  show  that  the  fast,  longitu- 
dinal or  (+  +  )  mode  is  highly  dispersive.  Given  the  geometry  of  two 
such  coupled  helices  and  the  relevant  data  on  an  electron  beam,  namely 
current,  voltage  and  beam  radius,  it  is  possible  to  arrive  at  an  estimate 
of  the  dependence  of  gain  on  frecjuency. 

Experiments  with  such  a  tube  showed  a  Ijandwidth  3.8  times  larger 
than  the  simple  estimate  would  show.  This  we  ascribe  to  the  presence 


COUPLED   HELICES  161 

of  the  dielectric  between  the  helices  in  the  actual  tube,  and  to  the  neglect 
of  power  propagated  in  the  form  of  spatial  harmonics. 

Nevertheless,  the  tube  operated  satisfactorily  with  distributed  non- 
reciprocal  ferrite  attenuation  along  the  whole  helix  and  gave,  at  the 
center  frequency  of  4,500  mc/s  more  than  40  db  stable  gain. 

The  gain  fell  to  zero  at  3,950  mc/s  at  one  end  of  the  band  and  at 
4,980  mc/s  at  the  other.  The  forward  loss  was  12  db.  The  backward 
loss  was  of  the  order  of  50  db  at  the  maximum  gain  frequency. 

3.4  Devices  Using  Both  Modes 

In  this  section  we  shall  discuss  applications  of  the  coupled-helix  princi- 
ple which  depend  for  their  function  on  the  simultaneous  presence  of  both 
the  transverse  and  the  longitudinal  modes.  When  present  in  substantially 
equal  magnitude  a  spatial  beat-phenomenon  takes  place,  that  is,  RF 
power  transfers  back  and  forth  between  inner  and  outer  helix. 

Thus,  there  are  points,  periodic  with  distance  along  each  helix,  where 
there  is  substantially  no  current  or  voltage;  at  these  points  a  helix  can  be 
terminated,  cut-off,  or  connected  to  external  circuits  without  detriment. 

The  main  object,  then,  of  all  devices  discussed  in  this  section  is  power 
transfer  from  one  helix  to  the  other;  and,  as  will  be  seen,  this  can  be  ac- 
complished in  a  remarkably  efficient,  elegant,  and  broad-band  manner. 

3.4.1  Coupled-Helix  Transducer 

It  is,  by  now,  a  well  known  fact  that  a  good  match  can  be  obtained 
between  a  coaxial  line  and  a  helix  of  proportions  such  as  used  in  TWT's.  A 
wire  helix  in  free  space  has  an  effective  impedance  of  the  order  of  100 
ohms.  A  conducting  shield  near  the  helix,  however,  tends  to  reduce  the 
helix  impedance,  and  a  value  of  70  or  even  50  ohms  is  easily  attained. 
Pro\'ided  that  the  transition  region  between  the  coaxial  line  and  the 
helix  does  not  present  too  abrupt  a  change  in  geometry  or  impedance, 
relatively  good  transitions,  operable  over  bandwidths  of  several  octaves, 
can  l)e  made,  and  are  used  in  practice  to  feed  into  and  out  of  tubes  em- 
ploying helices  such  as  TWT's  and  backward-wave  oscillators. 

One  particularly  awkward  point  remains,  namely,  the  necessity  to  lead 
the  coaxial  line  through  the  tube  envelope.  This  is  a  complication  in 
manufacture  and  reciuires  careful  positioning  and  dimensioning  of  the 
helix  and  other  tube  parts. 

Coupled  helices  offer  an  opportunity  to  overcome  this  difficulty  in  the 
form  of  the  so-called  coupled-helix  transducer,  a  sketch  of  which  is 
shown  in  Fig.  3.2.  As  has  been  shown  in  Section  2.3,  with  helices  having 


162 


THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


the  same  velocity  an  overlap  of  one  half  of  a  beat  wavelength  will  result 
in  a  100  per  cent  power  transfer  from  one  helix  to  the  other.  A  signal  in- 
troduced into  the  outer  helix  at  point  A  by  means  of  the  coaxial  line  will 
be  all  on  the  inner  helix  at  point  B,  nothing  remaining  on  the  outer  helix. 
At  that  point  the  outer  helix  can  be  discontinued,  or  cut  off;  since  there 
is  no  power  there,  the  seemingly  violent  discontinuity  represented  by  the 
'open"  end  of  the  helix  will  cause  no  reflection  of  power.  In  practice,  un- 
fortunately, there  are  always  imperfections  to  consider,  and  there  will 
often  be  some  power  left  at  the  end  of  the  coupler  helix.  Thus,  it  is  de- 
sirable to  terminate  the  outer  helix  at  this  point  non-reflectively,  as,  for 
instance,  by  a  resistive  element  of  the  right  value,  or  by  connecting  to  it 
another  matched  coaxial  line  which  in  turn  is  then  non-reflectively  ter- 
minated. 

It  will  be  seen,  therefore,  that  the  coupled-helix  transducer  can,  in 
principle,  be  made  into  an  efficient  device  for  coupling  RF  energy  from 
a  coaxial  line  to  a  helix  contained  in  a  dielectric  envelope  such  as  a  glass 
tube.  The  inner  helix  will  be  energized  predominantly  in  one  direction, 
namely,  the  one  away  from  the  input  connection.  Conversely,  energy 
traveling  initially  in  the  inner  helix  will  be  transferred  to  the  outer,  and 
made  available  as  output  in  the  respective  coaxial  line.  Such  a  coupled- 
helix  transducer  can  be  moved  along  the  tube,  if  required.  As  long  as  the 
outer  helix  completely  overlaps  the  inner,  operation  as  described  above 
should  be  assured.  By  this  means  a  new  flexibility  in  design,  operation 
and  adjustment  of  traveling-wave  tubes  is  obtained  which  could  not  be 
achieved  by  any  other  known  form  of  traveling-wave  tube  transducer. 
Naturally,  the  applications  of  the  coupled-helix  transducer  are  not 
restricted  to  TWT's  only,  nor  to  100  per  cent  power  transfer.  To  obtain 


Fig.  3.2  —  A  simple  coupled  helix  transducer. 


COUPLED    HELICES  1G3 

power  transfer  of  proportions  other  than  100  per  cent  two  possibilities 
are  open:  either  one  can  reduce  the  length  of  the  synchronous  coupling 
helix  appropriately,  or  one  can  deliberately  make  the  helices  non-syn- 
chronous. In  the  latter  case,  a  considerable  measure  of  broad-banding 
can  be  obtained  by  making  the  length  of  overlap  again  equal  to  one  half 
of  a  beat-wavelength,  while  the  fraction  of  power  transferred  is  deter- 
mined by  the  difference  of  the  helix  velocities  according  to  2.4.7.  An 
application  of  the  principle  of  the  coupled-helix  transducer  to  a  variable 
delay  line  has  been  described  by  L.  Stark  in  an  unpublished  memo- 
randum. 

Turning  again  to  the  complete  power  transfer  case,  we  may  ask: 
How  broad  is  such  a  coupler? 

In  Section  2.7  we  have  discussed  how  the  radial  falling-off  of  the  RF 
energy  near  a  helix  can  be  used  to  broad-band  coupled-helix  devices 
which  depend  on  relative  constancy  of  beat-wavelength  as  frequency 
is  varied.  On  the  assumption  that  there  exists  a  perfect  broad-band  match 
between  a  coaxial  line  and  a  helix,  one  can  calculate  the  performance  of 
a  coupled-helix  transducer  of  the  type  shown  in  Fig.  3.2. 

Let  us  define  a  center  frequency  co,  at  which  the  outer  helix  is  exactly 
one  half  beat-wavelength,  \b ,  long.  If  oj  is  the  frequency  of  minimum 
beat  wavelength  then  at  frequencies  coi  and  co2 ,  larger  and  smaller, 
respectively,  than  co,  the  outer  helix  will  be  a  fraction  5  shorter  than 
}i\b ,  (Section  2.7).  Let  a  voltage  amplitude,  Vo ,  exist  at  the  point  where 
the  outer  helix  is  joined  to  the  coaxial  line.  Then  the  magnitude  of  the 
voltage  at  the  other  end  of  the  outer  helix  will  be  |  F2  •  sin  (x5/2)  |  which 
means  that  the  power  has  not  been  completely  transferred  to  the  inner 
helix.  Let  us  assume  complete  reflection  at  this  end  of  the  outer  helix. 
Then  all  but  a  fraction  of  the  reflected  power  will  be  transferred  to  the 
inner  helix  in  a  reverse  direction.  Thus,  we  have  a  first  estimate  for  the 
"directivity"  defined  as  the  ratio  of  forward  to  backward  power  (in  db) 
introduced  into  the  inner  helix: 


D  = 


10  log  sin" 


(3.4.1.1) 


We  have  assumed  a  perfect  match  between  coaxial  line  and  outer  helix; 
thus  the  power  reflected  back  into  the  coaxial  line  is  proportional  to 
sin^(x5/2).  Thus  the  reflectivity  defined  as  the  ratio  of  reflected  to 
incident  power  is  given  in  db  by 


i^  =  10  log  sin'    ^  (3.4.1.2) 


164  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 

For  the  sake  of  definiteness,  let  us  choose  actual  figures:  let  /3a  =  2.0. 
and  hi  a  =  1.5.  And  let  us,  arbitrarily,  demand  that  R  always  be  less  than 
-20  db. 

This  gives  sin  (7r5/2)  <  0.316  and  7r5/2  <  18.42°  or  0.294  radians, 
8  <  0.205.  With  the  optimum  value  of  (Sea  =  1.47,  this  gives  the  mini- 
mum permissible  value  of  I3ca  of  1.47/(1  +  0.205)  =  1.22.  From  the 
graph  on  Fig.  2.2  this  corresponds  to  values  of  jSa  of  1.00  and  3.50. 
Therefore,  the  reflected  power  is  down  20  db  over  a  frequency  range  of 
aj2/aji  =  3,5  to  one.  Over  the  same  range,  the  directivity  is  better  than 
10  to  one.  Suppose  a  directivity  of  better  than  20  db  were  required. 
This  requires  sin  (7r5/2)  =  0.10,  8  =  0.0638  and  is  obtained  over  a  fre- 
quency range  of  approximately  two  to  one.  Over  the  same  range,  the 
reflected  power  would  be  down  by  40  db. 

In  the  above  example  the  full  bandwidth  possibilities  have  not  been 
used  since  the  coupler  has  been  assumed  to  have  optimum  length  when 
jSctt  is  maximum.  If  the  coupler  is  made  longer  so  that  when  I3ca  is  maxi- 
mum it  is  electrically  short  of  optimum  to  the  extent  permissible  by 
the  quality  requirements,  then  the  minimum  allowable  (S^a  becomes  even 
smaller.  Thus,  for  h/a  =1.5  and  directivity  20  db  or  greater  the  rea- 
lizable bandwidth  is  nearly  three  to  one. 

When  the  coupling  helix  is  non-reflectively  terminated  at  both  ends, 
either  by  means  of  two  coaxial  lines  or  a  coaxial  line  at  one  end  and  a 
resistive  element  at  the  other,  the  directivity  is,  ideally,  infinite,  irrespec- 
tive of  frequency;  and,  similarly,  there  will  be  no  reflections.  The  power 
transfer  to  the  inner  helix  is  simply  proportional  to  cos  (t8/2).  Thus, 
under  the  conditions  chosen  for  the  example  given  above,  the  coupled- 
helix  transducer  can  approach  the  ideal  transducer  over  a  considerable 
range  of  frequencies. 

So  far,  we  have  inspected  the  performance  and  bandwith  of  the 
coupled-helix  transducer  from  the  most  optimistic  theoretical  point  of 
view.  Although  a  more  realistic  approach  does  not  change  the  essence 
of  our  conclusions,  it  does  modify  them.  For  instance,  we  have  neglected 
dispersion  on  the  helices.  Dispersion  tends  to  reduce  the  maximum  at- 
tainable bandwidth  as  can  be  seen  if  Fig.  2.4.2  rather  than  Fig.  2.2  is 
used  in  the  example  cited  above.  The  dielectric  that  exists  in  the  annular 
region  between  coupled  concentric  helices  in  most  practical  couplers 
may  also  affect  the  bandwidth. 

In  practice,  the  performance^  of  coupled-hc^lix  transducers  has  been 
short  of  the  ideal.  In  the  first  place,  the  match  from  a  coaxial  line  to  a 
helix  is  not  perfect.  Secondly,  a  not  inappreciable  fraction  of  the  RF 
power  on  a  real  wire  helix  is  propagated  in  the  form  of  spatial  harmonic 


COUPLED   HELICES 


165 


28 


26 


24 


22 


20 


18 


)6 

in 

_i 

LU 

m  (4 
u 


12 


10 


r\ 

\ 

\ 
\ 

\ 

'   *  / 
'    *  / 
1     t  / 

[\ 

n 

[  1 

I 

1      1 

\j 

^ 

\ 

Wf 

\ 

1 

\ 

\ 

I      / 
I    / 
\  / 

\1^ 

U~ 

/ 

/ 
/ 

\ 

.' 
1 

A 

\J 

\- 

/  \ 

\ 

A 

/ 

Vi 

\ 
\ 

\ 

1 

1 

/ 
1 

p 

OUPLER   DIRECTIVITY 
ETURN   LOSS 

\ 

\ 

1 

J 

\ 

A 

V 

I 

/ 

l 

1.5 


2.5  3  4 

FREQUENCY   IN    KILOMEGACYCLES 


Fig.  3.3  — •  The  return  loss  and  directivity  of  an  experimental  100  per  cent 
coupled-helix  transducer. 

wave  components  which  have  variations  with  angle  around  the  helix- 
axis,  and  coupling  between  such  components  on  two  helices  wound  in 
opposite  directions  must  be  small.  Finally,  there  are  the  inevitable  me- 
chanical inaccuracies  and  misalignments. 

Fig.  3.3  shows  the  results  of  measurements  on  a  coupled-helix  trans- 
ducer with  no  termination  at  the  far  end. 


3.4.2  Coupled-Helix  Attenuator 

In  most  TWT's  the  need  arises  for  a  region  of  heavy  attenuation 
somewhere  between  input  and  output;  this  serves  to  isolate  input  and 
output,  and  prevents  oscillations  due  to  feedback  along  the  circuit.  Be- 
cause of  the  large  bandwidth  over  which  most  TWT's  are  inherently 
capable  of  amplifying,  substantial  attenuation,  say  at  least  60  db,  is 


166  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

required  over  a  bandwidth  of  maybe  2  octaves,  or  even  more.  Further- 
more, such  attenuation  should  present  a  very  good  match  to  a  wave  on 
the  heHx,  particularly  to  a  wave  traveling  backwards  from  the  output 
of  the  tube  since  such  a  wave  will  be  amplified  by  the  output  section  of 
the  tube. 

Another  requirement  is  that  the  attenuator  should  be  physically  as 
short  as  possible  so  as  not  to  increase  the  length  of  the  tube  unneces- 
sarily. 

Finally,  such  attenuation  might,  with  advantage,  be  made  movable 
during  the  operation  of  the  tube  in  order  to  obtain  optimum  performance, 
perhaps  in  respect  of  power  output,  or  linearity,  or  some  other  aspect. 

Coupled-helix  attenuators  promise  to  perform  these  functions  satis- 
factorily. 

A  length  of  outer  helix  (synchronous  with  the  inner  helix)  one  half  of  a 
beat  wavelength  long,  terminated  at  either  end  non-reflectively,  forms  a 
very  simple,  short,  and  elegant  solution  of  the  coupled-helix  attenuator 
problem.  A  notable  weakness  of  this  form  of  attenuator  is  its  relatively 
narrow  bandwidth.  Proceeding,  as  before,  on  the  assumption  that  the 
attenuator  is  a  fraction  8  larger  or  smaller  than  half  a  beat  wavelength 
at  frequencies  coi  and  W2  on  either  side  of  the  center  frequency  co,  we  find 
that  the  fraction  of  power  transferred  from  the  inner  helix  to  the  attenu- 
ator is  then  given  by  (1  —  sin"  (ir8/2)).  The  attenuation  is  thus  simply 

A  =  sin^  (I) 

For  helices  of  the  same  proportions  as  used  before  in  Section  3.4.1,  we 
find  that  this  will  give  an  attenuation  of  at  least  20  db  over  a  frequency 
band  of  two  to  one.  At  the  center  frequency,  coo ,  the  attenuation  is  in- 
finite; —  in  theory. 

Thus  to  get  higher  attenuation,  it  would  be  necessary  to  arrange  for  a 
sufficient  number  of  such  attenuators  in  tandem  along  the  TWT.  More- 
over, by  properly  staggering  their  lengths  within  certain  ranges  a  wdder 
attenuation  band  may  be  achieved.  The  success  of  such  a  scheme  largely 
depends  on  the  ability  to  terminate  the  helix  ends  non-reflectively.  Con- 
siderable work  has  been  done  in  this  direction,  but  complete  success  is 
not  yet  in  sight. 

Another  basically  different  scheme  for  a  coupled-helix  attenuator  rests 
on  the  use  of  distributed  attenuation  along  the  coupling  helix.  The  diffi- 
culty with  any  such  scheme  lies  in  the  fact  that  unequal  attenuation  in 
the  two  coupled  helices  reduces  the  coupling  between  them  and  the  moi'c 
they  differ  in  respect  to  attenuation,  the  less  the  coupling.  Naturally,  one 


COUPLED   HELICES  167 

would  wish  to  have  as  Httle  attenuation  as  practicable  associated  with 
the  inner  helix  (inside  the  TWT).  This  requires  the  attenuating  element 
to  be  associated  with  the  outer  helix.  Miller  has  shown  that  the  maxi- 
mum total  power  reduction  in  coupled  transmission  systems  is  obtained 
when 

ai  —  0:2 


where  ai  and  012  are  the  attenuation  constants  in  the  respective  systems, 
and  ^b  the  beat  phase  constant.  If  the  inner  helix  is  assumed  to  be  loss- 
less, the  attenuation  constant  of  the  outer  helix  has  to  be  effectively  equal 
to  the  beat  wave  phase  constant.  It  turns  out  that  60  db  of  attenuation 
requires  about  3  beat  wavelengths  (in  practice  10  to  20  helix  wave- 
lengths). The  total  length  of  a  typical  TWT  is  only  3  or  4  times  that, 
and  it  will  be  seen,  therefore,  that  this  scheme  may  not  be  practical  as 
the  only  means  of  providing  loss. 

Experiments  carried  out  Avith  outer  helices  of  various  resistivities  and 
thicknesses  by  K.  M.  Poole  (then  at  the  Clarendon  Laboratory,  Oxford, 
England)  tend  to  confirm  this  conclusion.  P.  D.  Lacy"  has  described  a 
coupled  helix  attenuator  which  uses  a  multifilar  helix  of  resistance 
material  together  with  a  resistive  sheath  between  the  helices. 

Experiments  were  performed  at  Bell  Telephone  Laboratories  with  a 
TWT  using  a  resistive  sheath  (graphite  on  paper)  placed  between  the 
outer  helix  and  the  quartz  tube  enclosing  the  inner  helix.  The  attenua- 
tions were  found  to  be  somewhat  less  than  estimated  theoretically.  The 
attenuator  helix  was  movable  in  the  axial  direction  and  it  w^as  instructive 
to  observe  the  influence  of  attenuator  position  on  the  power  output  from 
the  tube,  particularly  at  the  highest  attainable  power  level.  As  one  might 
expect,  as  the  power  level  is  raised,  the  attenuator  has  to  be  moA-ed  nearer 
to  the  input  end  of  the  tube  in  order  to  obtain  maximum  gain  and  power 
output.  In  the  limit,  the  attenuator  helix  has  to  be  placed  right  close  to 
the  input  end,  a  position  which  does  not  coincide  with  that  for  maximum 
low-level  signal  gain.  Thus,  the  potential  usefulness  of  the  feature  of 
mobility  of  coupled-helix  elements  has  been  demonstrated. 

4.  CONCLUSION 

In  this  paper  we  have  made  an  attempt  to  develop  and  collect  together 
a  considerable  body  of  information,  partly  in  the  form  of  equations, 
partl}^  in  the  form  of  graphs,  which  should  be  of  some  help  to  workers 
in  the  field  of  microwave  tubes  and  devices.  Because  of  the  crudity  of  the 
assumptions,  precise  agreement  between  theory  and  experiment  has  not 


168  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 

been  att-aiiu>(l  iiur  can  it  l)c  expected.  Nevertheless,  the  kind  of  physical 
phenomena  occurring  with  coupled  helices  are,  at  least,  qualitatively 
described  here  and  should  permit  one  to  develop  and  construct  various 
types  of  (lexices  with  fair  chance  of  success. 

ACKNOWLEDGEMENTS 

As  a  final  note  the  authors  wish  to  express  their  appreciation  for  the 
patient  work  of  Mrs.  C.  A.  Lambert  in  computing  the  curves,  and  to 
G.  E.  Korb  for  taking  the  experimental  data. 

Appendix  i 
i.  solution  of  field  equations 

In  this  section  there  is  presented  the  field  equations  for  a  transmission 
system  consisting  of  two  helices  aligned  with  a  common  axis.  The  propa- 
gation properties  and  impedance  of  such  a  transmission  system  are  dis- 
cussed for  various  ratios  of  the  outer  helix  radius  to  the  inner  helix  radius. 
This  system  is  capable  of  propagating  two  modes  and  as  previously 
pointed  out  one  mode  is  characterized  by  a  longitudinal  field  midway 
between  the  two  helices  and  the  other  is  characterized  by  a  transverse 
field  midway  between  the  tw^o  helices. 

The  model  which  is  to  be  treated  and  shown  in  Fig.  2.3  consists  of  an 
inner  helix  of  radius  a  and  pitch  angle  \pi  which  is  coaxial  with  the  outer 
helix  of  radius  6  and  pitch  angle  \j/2 .  The  sheath  helix  model  will  be 
treated,  wherein  it  is  assumed  that  helices  consist  of  infinitely  thin  sheaths 
which  allow  for  ciuTent  flow-  only  in  the  direction  of  the  pitch  angle  \p. 

The  components  of  the  field  in  the  region  inside  the  inner  helix,  be- 
tween the  two  helices  and  outside  the  outer  helix  can  be  written  as 
follows  —  inside  the  inner  helix 

H,,  =  BrIoM  (1) 

E.,  =  B^hM  (2) 

H,,  =  j  -  BMyr)  (3) 

7 

Hr,  =  ^^  BMyr)  (4) 

7 

E,,  =   -j  "^  BMyr)  (5) 

7 

Er,  =  -^  BJ,(rr)  (()) 

7 


COUPLED   HELICES  169 

and  between  the  two  helices 

H,,  =  BMrr)  +  BJuirr)  '  (7) 

E.,  =  BJoiyr)  +  B^oiyr)  (8) 

H,,  =  ^~  [B,h(yr)  -  B^^(yr)]  (9) 

7 

Hr,  =  -^  [53/1(7/0  -  BJuiyr)]  (10) 

7 

E,,  =    -  J  ^  [B^hiyr)  -  BJuiyr)]  (11) 

7 

Er,  =  -^  [BMyr)  -  BJv,{yr)\  (12) 

7 

and  outside  the  outer  hehx 

H.^  =  B,Ko(yr)  (13) 

E,,  =  58/vo(7r)  (14) 

^.s  =   -J-  BsK,{yr)  (15) 

7 

Hr,  =  ^^  5,Ki(7r)  (16) 

7 

^,,  =  i  —  BJuiyr)  (17) 

7 

^r«  =    ^^  58Ki(7r)  (18) 

7 

With  the  sheath  helix  model  of  current  flow  only  in  the  direction  of  wires 
we  can  specify  the  usual  boundary  conditions  that  at  the  inner  and  outer 
helix  radius  the  tangential  electric  field  must  be  continuous  and  per- 
pendicular to  the  wires,  whereas  the  tangential  component  of  magnetic 
field  parallel  to  the  current  flow  must  be  continuous.  These  can  be  written 
as 

E,  sin  t/'  +  E^  cos  ^  =  0  (19) 

'  E, ,  E^  and  (H,  sin  \f/  -f  H^  cos  \p)  be  equal  on  either  side  of  the  helix. 
By  applying  these  conditions  to  the  two  helices  the  following  equations 
are  obtained  for  the  various  coefficients. 


170  THE    BELL    SYSTEM    TECHNICAL   JOURNAL,    JANUARY    1956 


First,  we  will  define  a  more  simple  set  of  parameters.  We  will  denote 

Io(ya)  by  /oi        and        h{yh)  by  /02 ,  etc. 

Further  let  us  use  the  notation  introduced  by  Humphrey,  Kite  and 
James"  in  his  treatment  of  coaxial  helices. 


Poi  ^  laiKoi  P02  =  ToiKa2  Rq  =  I01K02 

Pn  =  InKn  P12  =  InKu  Ri  =  /iii^i2 

and  define  a  common  factor  (C.F.)  by  the  equation 

r(/3oa  cot  hY  p   p  (/3oa  cot  ^pif  cot  i/'z  „  r, 

\_       (yay  {jay  cot  t^i 

+  Ro'  —  PoiP 


(20) 


.,] 


(21) 


With  all  of  this  we  can  now  write  for  the  coefficients  of  equations  1 
through  18: 


y   ju  j8oa  cot  \pi  1 02 

U  iQoa  cot  1^1  7oi/vi2  RiSoa  cot  i^i) 

y  M  ""to     C.F.  L 

^4  _    _  •       / £_  /3oa  cot  1^1  /pi/ii  r( 

B^~      -^  T   M        7^        C.F.  L"      (7a)^ 


5 
5 


(7a)'^ 
(/3oa  cot  1^2)^ 


cot  1A2   p 
cot  ;^i        J 

P12  —   jPo2 


■] 


B5 

B, 
Bt 


Ro 
C.F. 


Ro  — 


((Soa  cot  xl/iY  cot  1/' 


(7a^) 
(/3oa  cot  1^2) 


cot  l/' 


;«'] 


(7a)^ 


12  —  -P02 


B7  _    •    .  /£  i3oa  cot  lAi     1      /oi  r 
5;  ~  "^  y  M        7a        C.F.  K12  L 


Bs  _    (|8oa  cot  i/'i)"  cot  1/^2      /pi       "" 
B2  {yay        coT^i  C.F.Po 


P02R1  — 
P02R1  - 


cot  l/'2 
cot  i/'i 

cot  l// 
cot  \l/ 


2R0 
-  P12R0 


(22) 
(23) 
(24) 
(25) 
(26) 
(27) 
(28) 


The  last  equation  necessary  for  the  solution  of  our  field  problem  is  the 
transcendental  equation  for  the  propagation  constant,  7,  which  can  be 


COUPLED    HELICES 


171 


written 


Ro  — 


(i8o  a  cot  \J/iY  cot  ^2  „ 
(yaY        cot  4/1 


[ 


=        P02   - 


(jSo  a  cot  \p2)    D 

? Vi -^   12 


Poi  - 


(/3oa  cot  ^0" 
(yay 


_     (29) 


11 


The  solutions  of  this  equation  are  plotted  in  Fig.  4.1. 

There  it  is  seen  that  there  are  two  values  of  7,  one,  yt  ,  denoting  the 
slow  mode  with  transverse  fields  between  helices  and  the  other,  yt , 
denoting  the  fast  mode  with  longitudinal  fields  midway  between  the  two 
helices. 


5.0 


4.S 


4.0 


3.5 


3.0 


ra 


2.6 


2.0 


1.5 


1.0 


0.5 


4  =  1.25 

// 

/ 

COT  5^2 
COT  ^1 

0.82 

0.90 

0.98 

^ 

#■ 

/t 

// 

A 

na 

/ 

f 

A 

y 

f 

/ 

r 

/ 

I 

/ 

if 

< 

A 

/ 

-<^^ 

^ 

•y 

L 

-- 

•=**^ 

0.5  1.0  1.6 


2.0  2.5  3.0 

/3o  a  COT  yj 


3.5  4.0  4.5  5.0 


Fig.  4.1.1 —-The  radial  propagation  constants  associated  with  the  transverse 
and  longitudinal  modes  on  coupled  coaxial  sheath  helices  given  as  a  function  of 
|3oa  cot  ^i-i  for  several  values  of  hja  =  1.25. 


172  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


These  equations  can  now  be  used  to  compute  the  power  flow  as  defined 
by 

P  =  }4  Re  j  E  XH' 
which  can  be  written  in  the  form 


dA 


(30) 


r^;^(o)T 
L  ^'p  J 


fo  ©    ^^-'  ''' 


(31) 


where 


[F{ya,  yb)]     = 


(( 


W  + 


(i8oa  cot  i/' 
{yar 


^  /n^) 


(In'  -  /oi/2i)(C.F.)- 


-      A'02'  + 


240  (C.F.)' 

(i8oa  cot  1/^1)' 
(t«)'^ 


/or/n-  r 


(80a  cot  ;^iY 


' 


/Vl2"         i^O    - 


ya 

((Soft  cot  i^i)'  cot  \p2 
(ya)''        cot  i/'i 


Rx 


- )  (/02/22  —  /12')  4"  (/ii   —  /01/21) 


,    /p         (/3oa  cot  1A1)-  cot  \i/2  p  Wp  (^0^  ^'0^'  "^2)''  p 


(ya)'^        cot  i/'i 


(7a)^ 


(  - )  i'lInKu  +  /02/V22  +  /22X02)  —  (2/iiKii  +  /01K21  +  /21/voi) 

ot  ^2)'^  p  T 


(32) 


2     ,     (^ofl  cot  l/'i)     J    ■> 
•'01    i-  7 r;; ^11 


(l3oa  cot 


-  I    (K02K22  —  K12 )  —  (K01K21  —  Kn) 


.a, 


+ 


(/3oa  cot  i^i)"  A^ 


■     2    ,    (/Soa  cot  i/'2)"  J  2  J.  2 


(7a)'^ 


cot  1/^2   p     J. 
I    02itl    —    -— r-    i    12A0 

cot  1^1 


[/Vo2A'22    —    /V12"] 


In  (32)  we  find  the  power  in  the  transverse  mode  by  using  values  of 


COUPLED   HELICES 


173 


5.0 


0.5 


2.0  2.5  3.0 

/3o  a  COT  y/ 


5.0 


Fig.  4.1.2  —  The  radial  propagation  constants  associated  with  the  transverse 
and  longitudinal  modes  on  coupled  coaxial  sheath  helices  given  as  a  function  of 
^ofl  cot  \}/i  when  h/a  —  1.50. 

yt  obtained  from  (29)  and  similarly  the  power  in  the  longittidinal  mode  is 
found  by  using  values  of  yi . 


II.   FINDING  r 

When  coaxial  helices  are  used  in  a  transverse  field  amplifier,  only  the 
transverse  field  mode  is  of  interest  and  it  is  important  that  the  helix 
parameters  be  adjusted  such  that  there  is  no  longitudinal  field  at  some 
radius,  f,  where  the  cylindrical  electron  beam  will  be  located.  This  condi- 
tion can  be  expressed  by  equating  Ez  to  zero  at  r  =  f  and  from  (8) 


BMyr)  +  B^,{yf)  =  0 


(33) 


174  THE   BELL    SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 

which  can  be  written  with  (25)  and  (26)  as 
(jSott  cot  ipiY  cot  \f/2 


K(i2    Ri 


[ 


02      ilO 


(7a)-         cot  \{/i 
=  /oi 


Ri 


loM 


(/3oa  cot  \l/2)- 

■I  02  —  7 rr, rn 


(34) 


Koiyf) 


This  equation  together  with  (29)  enables  one  to  evaluate  f/a  versus  j8oa 
cot  \l/i  for  various  ratios  of  b/a  and  cot  i^2/cot  xpi  .  The  results  of  these 
calculations  are  shown  in  Fig.  3.1. 


5.0 


4.5 


4.0 


3.5 


3.0 


7a 


2.5 


2,0 


0.5 


Fig.  4.1. .3  —  The  radial  propagation  constants  associated  with  the  transverse 
and  longitudinal  modes  on  coupled  coaxial  sheath  helices  given  as  a  finiclion  of 
0oa  cot  \{/i  when  b/a  =  1.75. 


i 


COUPLED   HELICES 


175 


5.0 


7a 


2.0  2.5  3.0 

/Oo  <3  COT  ^, 


3.5 


4.0 


4.5 


5.0 


Fig.  4.1.4  —  The  radial  propagation  constants  associated  with  the  transverse 
and  longitudinal  modes  on  coupled  coaxial  sheath  helices  given  as  a  function  of 
/3oa  cot  yp\  when  6/a  =  2.0. 


III.    COMPLETE   POWER  TRANSFER 

For  coupled  heli.x  applications  we  require  the  coupled  helix  parame- 
ters to  be  adjusted  so  that  RF  power  fed  into  one  helix  alone  will  set  up 
the  transverse  and  longitudinal  modes  equal  in  amplitude.  For  this 
condition  the  power  from  the  outer  helix  will  transfer  completely  to  the 
inner  helix.  The  total  current  density  can  be  written  as  the  sum  of  the 
current  in  the  longitudinal  mode  and  the  transverse  mode.  Thus  for  the 
inner  helix  we  have 


-i&li 


J  a   =    Jate-''''  +    Jate 


.-J^<2 


(35) 


17G  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


7?,     2.5 


Fig.  4.1.5  —  The  radial  propagation  constants  associated  with  the  transverse 
and  longitudinal  modes  on  coupled  coa.xial  sheath  helices  given  as  a  function  of 

/3oa  cot  i/-!  when  hi  a  =  3.0. 


and  for  the  outer  helix 
For  complete  power  transfer  we  ask  that 

•J  hi    —    J  hi 

when  Jo  is  zero  at  the  input  {z  =  0) 
or 

Jbt    _  Jbt 

J  at  J  at 


\ 


(36)    \ 


(37) 


COUPLED   HELICES  177 

Now  J  at  is  equal  to  the  discontinuity  in  the  tangential  component  of 
magnetic  field  which  can  be  written  at  r  =  a 

J  at  =  {H,z  cos  ^i  —  //^5  sin  \pi)  —  (H,i  cos  i/'i  -  H^o  sin  \f/i) 

\^'hich  can  be  written  as 

Ja(  =  -  (H,i  -  H,3)a((cot  i/'i  +  tau  xj/i)  slu  \Pi  (38) 

and  similarily  at  r  =  h 

Jb(  =  —  (H^7  —  H,s)b({cot  \p2  +  tan  4^2)  sin  i/'2  (39) 

Equations  (38)  and  (39)  can  be  combined  with  (37)  to  give  as  the  condi- 
tion for  complete  power  transfer 

At  =  -At  (40) 

where 

^  =  V (yay  /  ni) 

(T    J^    _i-  r   V   \(  T?         (/3oa  cot  <Ai)'^  cot  1^2  „  \ 
\  {yo,y      cot  i/'i     / 

In  (40)  At  is  obtained  by  substituting  jt  into  (41)  and  At  is  obtained  by 
substituting  7  <  into  (41). 

The  value  of  cot  i/'o/cot  i/'i  necessary  to  satisfy  (40)  is  plotted  in  Fig. 
2.8. 

In  addition  to  cot  i/'o/cot  i/'i  it  is  necessary  to  determine  the  interference 
wavelength  on  the  helices  and  this  can  be  readily  evaluated  by  consider- 
ing (36)  which  can  now  be  written 

or 

/,  =  /,,.-«^'+^''-''^>  cos  ^ilJZ^  ,  (48) 

and 

J,  =  J.ce-'''^'^'^''"'  cos  M/3i^  (49) 

where  we  have  defined 

iSfcO  =  {yta  —  jta)  (50) 

This  value  of  /S^  is  plotted  versus  /3oa  cot  i/'i  in  Fig.  2.4. 


178  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

BIBLIOGRAPHY 

1.  J.  R.  Pierce,  Traveling  Wave  Tubes,  p.  44,  Van  Nostrand,  1950. 

2.  R.  Kompfner,  Experiments  on  Coupled  Helices,  A.  E.  R.  E.  Report  No. 

G/M98,  Sept.,  1951. 

3.  R.  Kompfner,  Coupled  Helices,  paper  presented  at  I.  R.  E.  Electron  Tube 

Conference,  1953,  Stanford,  Cal. 

4.  G.  Wade  and  N.  Rynn,  Coupled  Helices  for  Use  in  Traveling-Wave  Tubes, 

I.R.E.  Trans,  on  Electron  Devices,  Vol.  ED-2,  p.  15,  July,  1955. 

5.  S.  E.  Miller,  Coupled  Wave  Theory  and  Waveguide  Applications,  B.S.T.J., 

33,  pp.  677-693,  1954. 

6.  M.  Chodorow  and  E.  L.  Chu,  The  Propagation  Properties  of  Cross-Wound 

Twin  Helices  Suitable  for  Traveling-Wave  Tubes,  paper  presented  at  the 
Electron  Tube  Res.  Conf.,  Stanford  Univ.,  June,  1953. 

7.  G.  M.  Branch,  A  New  Slow  Wave  Structure  for  Traveling-Wave  Tubes,  paper 

presented  at  the  Electron  Tube  Res.  Conf.,  Stanford  Univ.,  June,  1953. 
G.  M.  Branch,  E.xperimental  Observation  of  the  Properties  of  Double  Helix 
Traveling-Wave  Tubes,  paper  presented  at  the  Electron  Tube  Res.  Conf., 
Univ.  of  Maine,  June,  1954. 

8.  J.  S.  Cook,  Tapered  Velocity  Couplers,  B.S.T.J.  34,  p.  807,  1955. 

9.  A.  G.  Fox,  Wave  Coupling  by  Warped  Normal  Modes,  B.S.T.J.,  34,  p.  823, 

1955. 

10.  W.  H.  Louisell,  Analysis  of  the  Single  Tapered  Mode  Coupler,  B.S.T.J.,  34, 

p.  853. 

11.  B.  L.  Humphrey's,  L.  V.  Kite,  E.  G.  James,  The  Phase  Velocity  of  Waves  in  a 

Double  Helix,  Report  No.  9507,  Research  Lab.  of  G.E.C.,  England,  Sept., 
1948. 

12.  L.  Stark,  A  Helical-Line  Phase  Shifter  for  Ultra-High  Frequencies,  Technical 

Report  No.  59,  Lincoln  Laboratory,  M.LT.,  Feb.,  1954. 

13.  P.  D.  Lacy,  Helix  Coupled  Traveling-Wave  Tube,  Electronics,  27,  No.  11, 

Nov..  1954. 


Statistical  Techniques  for  Reducing  the 
Experiment  Time  in  Reliability  Studies 

By  MILTON  SOBEL 

(Manuscript  received  September  19,  1955) 

Given  two  or  more  processes,  the  units  from  which  fail  in  accordance  with 
an  exponential  or  delayed  exponential  law,  the  problem  is  to  select  the  partic- 
ular process  with  the  smallest  failure  rate.  It  is  assumed  that  there  is  a  com- 
mon guarantee  period  of  zero  or  positive  duration  during  which  no  failures 
occur.  This  guarantee  period  may  be  known  or  unknown.  It  is  desired  to 
accomplish  the  above  goal  in  as  short  a  time  as  possible  without  invalidating 
certain  predetermined  probability  specifications.  Three  statistical  techniques 
are  considered  for  reducing  the  average  experiment  time  needed  to  reach  a 
decision. 

1 .  One  technique  is  to  increase  the  initial  number  of  units  put  on  test. 
This  technique  will  substantially  shorten  the  average  experiment  time.  Its 
effect  on  the  probability  of  a  correct  selection  is  generally  negligible  and  in 
some  cases  there  is  no  effect. 

2.  Another  technique  is  to  replace  each  failure  immediately  by  a  new 
unit  from  the  same  process.  This  replacement  technique  adds  to  the  book- 
keeping of  the  test,  but  if  any  of  the  population  variances  is  large  (say  in 
comparison  with  the  guarantee  period)  then  this  technique  will  result  in  a 
substantial  saving  in  the  average  experiment  time. 

3.  A  third  technique  is  to  use  an  appropriate  sequential  procedure.  In 
many  problems  the  sequential  procedure  results  in  a  smaller  average  experi- 
ment time  than  the  best  non-sequential  procedure  regardless  of  the  true 
failure  rates.  The  amount  of  saving  depends  principally  on  the  ^'distance'" 
between  the  smallest  and  second  smallest  failure  rates. 

For  the  special  case  of  two  processes,  tables  are  given  to  show  the  proba- 
bility of  a  correct  selection  and  the  average  experiment  time  for  each  of  three 
types  of  procedures. 

Numerical  estimates  of  the  relative  efficiency  of  the  procedures  are  given 
by  computing  the  ratio  of  the  average  experiment  time  for  two  procedures  of 
different  type  with  the  same  initial  sample  size  and  satisfying  the  same 
probability  specification. 

179 


180  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

INTRODUCTION 

This  paper  is  concerned  with  a  study  of  the  advantages  and  disad- 
vantages of  three  statistical  techniques  for  reducing  the  average  dura- 
tion of  hfe  tests.  These  techniques  are: 

1.  Increasing  the  initial  number  of  units  on  test. 

2.  Using  a  replacement  technique. 

3.  Using  a  sequential  procedure. 

To  show  the  advantages  of  each  of  these  techniques,  we  shall  consider 
the  problem  of  deciding  which  of  two  processes  has  the  smaller  failure 
rate.  Three  different  types  of  procedures  for  making  this  decision  will 
be  considered.  They  are: 

Ri ,     A  nonsequential,  nonreplacement  type  of  procedure 
E,2 ,     A  nonsequential,  replacement  type  of  procedure 
Rs ,     A  sequential,  replacement  type  of  procedure 
Within  each  type  wq  will  consider  different  values  of  n,  the  initial 
number  of  units  on  test  for  each  process.  The  effect  of  replacement  is 
shown  by  comparing  the  average  experiment  time  for  procedures  of 
type  1  and  2  with  the  same  value  of  n  and  comparable  probabilities  of  a 
correct  selection.  The  effect  of  using  a  sequential  rule  is  shown  by  com- 
paring the  average  experiment  time  for  procedures  of  type  2  and  3  with 
the  same  value  of  n  and  comparable  probabilities  of  a  correct  selection. 

ASSUMPTIONS 

1.  It  is  assumed  that  failure  is  clearly  defined  and  that  failures  are 
recognized  without  any  chance  of  error. 

2.  The  lifetime  of  individual  units  from  either  population  is  assumed 
to  follow  an  exponential  density  of  the  form 

f{x;  e,g)  =\  e-^^-")/"        iov  x  -^  g 

f(x;  e,g)  =  0  iorx<g 

where  the  location  parameter  g  ^  0  represents  the  common  guarantee 
period  and  the  scale  parameter  6  >  0  represents  the  unknown  parameter 
which  distinguishes  the  two  different  processes.  Let  Ox  ^  do  denote  the 
ordered  values  of  the  unknown  parameter  6  for  the  two  processes;  then 
the  ordered  failure  rates  are  given  by 

Xi  =    1/(01  +  {/)    ^  Xo  =    1/(02  -f  g)  (2) 

3.  It  is  not  known  which  process  has  the  parameter  di  and  which  has 
the  parameter  dt . 


REDUCING   TIME    IN   RELIABILITY   STUDIES  181 

4.  The  parameter  g  is  assumed  to  be  the  same  for  both  processes.  It 
may  be  known  or  unknown. 

5.  The  initial  number  n  of  units  put  on  test  is  the  same  for  both  pro- 
cesses. 

6.  All  units  have  independent  lifetimes,  i.e.,  the  test  environment  is 
not  such  that  the  failure  of  one  unit  results  in  the  failure  of  other  units 
on  test. 

7.  Replacements  used  in  the  test  are  assumed  to  come  from  the  same 
population  as  the  units  they  replace.  If  the  replacement  units  have  to 
sit  on  a  shelf  before  being  used  then  it  is  assumed  that  the  replacements 
are  not  affected  by  shelf-aging. 

CONCLUSIONS 

1.  Increasing  the  initial  sample  size  n  has  at  most  a  negligible  effect 
on  the  probability  of  a  correct  selection.  It  has  a  substantial  effect  on  the 
average  experiment  time  for  all  three  types  of  procedures.  If  the  value  of 
n  is  doubled,  then  the  average  time  is  reduced  to  a  value  less  than  or 
equal  to  half  of  its  original  value. 

2.  The  technique  of  replacement  always  reduces  the  average  experi- 
ment time.  This  reduction  is  substantial  when  ^  =  0  or  when  the  popu- 
lation variance  of  either  process  is  large  compared  to  the  value  of  g. 
This  decrease  in  average  experiment  time  must  always  be  weighed  against 
the  disadvantage  of  an  increase  in  bookkeeping  and  the  necessity  of 
having  the  replacement  units  available  for  use. 

3.  The  sequential  procedure  enables  the  experimenter  to  make  rational 
decisions  as  the  evidence  builds  up  without  waiting  for  a  predetermined 
number  of  failures.  It  has  a  shorter  average  experiment  time  than  non- 
sequential procedures  satisfying  the  same  specification.  This  reduction 
brought  about  by  the  sequential  procedure  increases  as  the  ratio  a  of 
the  two  failure  rates  increases.  In  addition  the  sequential  procedure 
always  terminates  with  a  decision  that  is  clfearly  convincing  on  the  basis 
of  the  observed  results,  i.e.,  the  a  posteriori  probability  of  a  correct 
selection  is  always  large  at  the  termination  of  the  experiment. 

SPECIFICATION    OF   THE   TEST 

Each  of  the  three  types  of  procedures  is  set  up  so  as  to  satisfy  the 
same  specification  described  below.  Let  a  denote  the  true  value  of  the 
ratio  61/62  which  by  definition  must  be  greater  than,  or  equal  to,  one. 
It  turns  out  that  in  each  type  of  procedure  the  probability  of  a  correct 
selection  depends  on  6i  and  62  only  through  their  ratio  a. 


182  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1950 

1.  The  experimenter  is  asked  to  specify  the  smallest  value  of  a  (say 
it  is  a*  >  I)  that  is  worth  detecting.  Then  the  interval  (1,  a*)  represents 
a  zone  of  indifference  such  that  if  the  true  ratio  a  lies  therein  then  we 
would  still  like  to  make  a  correct  selection,  but  the  loss  due  to  a  wrong 
selection  in  this  case  is  negligible. 

2.  The  experimenter  is  also  asked  to  specify  the  minimum  value  P*  > 
\'2  that  he  desires  for  the  probability  of  a  correct  selection  whenever 
a  ^  a*.  In  each  type  of  procedure  the  rules  are  set  up  so  that  the  proba- 
bility of  a  correct  selection  for  a  =  a*  is  as  close  to  P*  as  possible  without 
being  less  than  P*. 

The  two  constants  a*  >  1  and  \2  <  P*  <  1  are  the  only  quantities 
specified  by  the  experimenter.  Together  they  make  up  the  specification 
of  the  test  procedure. 

EFFICIENCY 

If  two  procedures  of  different  type  have  the  same  value  of  n  and  satisfy 
the  same  specification  then  we  shall  regard  them  as  comparable  and 
their  relative  efficiency  will  be  measured  by  the  ratio  of  their  average 
experiment  times.  This  ratio  is  a  function  of  the  true  a  but  we  shall 
consider  it  only  for  selected  values  of  a,  namely,  a  =  1,  a  =  a*  and 
a  =  CO . 

PROCEDURES  OF  TYPE  Ri  — •  NONSEQUENTIAL,  NONREPLACEMENT 

"The  same  number  n  of  units  are  put  on  test  for  each  of  the  two  pro- 
cesses. Experimentation  is  continued  until  either  one  of  the  two  samples 
produces  a  predetermined  number  r  (r  ^  n)  of  failures.  Experimenta- 
tion is  then  stopped  and  the  process  with  fewer  than  r  failures  is  chosen 
to  be  the  better  one." 


Table    I  —  Probability    of    a    Correct    Selection  —  Procedure 

Type  Ri 
(a  =  2,  any  g  '^  0,  to  be  used  to  obtain  r  for  a*  =  2) 


n 

r  =  1 

r  =  2 

r  =  3 

r  =  i 

1 

0.667 





. — . 

2 

0.667 

0.733 

— 

— 

3 

0.667 

0.738 

0.774 

— 

4 

0.667 

0.739 

0.784 

0.802 

10 

0.667 

0.741 

0.78!) 

0.825 

20 

0.667 

0.741 

0 .  790 

0.826 

00 

0.667 

0.741 

0.790 

0.827 

Note:  The  value  for  ?•  =  0  is  obviously  0.500  for  any  n. 


REDUCING   TIME   IN    RELIABILITY   STUDIES  183 

We  shall  assume  that  the  number  n  of  units  put  on  test  is  determined 
by  non -statistical  considerations  such  as  the  availability  of  units,  the 
availability  of  sockets,  etc.  Then  the  only  unspecified  number  in  the 
above  procedure  is  the  integer  r.  This  can  be  determined  from  a  table 
of  probabilities  of  a  correct  selection  to  satisfy  any  given  specification 
(a*,  P*).  If,  for  example,  a*  =  2  then  we  can  enter  Table  I.  If  n  is 
given  to  be  4  and  we  wish  to  meet  the  specification  a*  =  2,  P*  =  0.800 
then  we  would  enter  Table  I  with  n  —  4  and  select  r  =  4,  it  being  the 
smallest  value  for  which  P  ^  P*. 

The  table  above  shows  that  for  the  given  specification  we  would  also 
have  selected  r  =  4  for  any  value  of  n.  In  fact,  we  note  that  the  proba- 
bility of  a  correct  selection  depends  only  slightly  on  n.  The  given  value 
of  n  and  the  selected  value  of  r  then  determine  a  particular  procedure 
of  type  Ri ,  say,  Ri(n,  r). 

The  average  experiment  time  for  each  of  several  procedures  R\{n,  r) 
is  given  in  Table  II  for  the  three  critical  values  of  the  true  ratio  a, 
namely,  a  =  \,  a  =  a*  and  a  =  oo .  Each  of  the  entries  has  to  be  multi- 
plied by  6-1 ,  the  smaller  of  the  two  d  values,  and  added  to  the  common 
guarantee  period  g.  For  n  =  oo  the  entry  should  be  zero  (-\-g)  but  it 
was  found  convenient  to  put  in  place  of  zero  the  leading  term  in  the 
asymptotic  expansion  of  the  expectation  in  powers  of  I/71.  Hence  the 
entry  for  n  =   00  can  be  used  for  any  large  n,  say,  n  ^  25  when  r  ^  4. 

We  note  in  Table  II  the  undesirable  feature  that  for  each  procedure 
the  average  experiment  time  increases  with  a  for  fixed  62  .  For  the  se- 
quential procedure  we  shall  see  later  that  the  average  experiment  time 
is  greater  at  a  =  a*  than  at  either  a  =  1  or  a  =  00 .  This  is  intuitively 
more  desirable  since  it  means  that  the  procedure  spends  more  time  when 
the  choice  is  more  difficult  to  make  and  less  time  when  we  are  indifferent 
or  when  the  choice  is  easy  to  make. 

PROCEDURES  OF  TYPE  R2  —  NONSEQUENTIAL,  REPLACEMENT 

"Such  procedures  are  carried  out  exactly  as  for  procedures  oiRi  except 
that  failures  are  immediately  replaced  by  new  units  from  the  same 
population." 

To  determine  the  appropriate  value  of  r  for  the  specification  a*  =  2, 
P*  =  0.800  when  g  =  0  we  use  the  last  row  of  Table  I,  i.e.,  the  row 
marked  n  =  ^ ,  and  select  r  =  4.  The  probability  of  a  correct  selection 
for  procedures  of  type  Ro  is  exactly  the  same  for  all  values  of  n  and  de- 
pends only  on  r.  Furthermore,  it  agrees  wdth  the  probability  for  pro- 
cedures of  type  Ri  with  n  =  co  so  that  it  is  not  necessary  to  prepare  a 
separate  table. 


PL, 


II 


H 

PM 

H 

K 
P 

o 

0  .^ 
Pi     d 

1  '^ 
I  «3 


'a 


2  S 


CC  Cl  t--  o 
00  r^  r-i  o 

O  '^  (N  o 


^  Ci  o  o 
»o  CO  o  CO 

00  ^  (M  CD 

.-I  o  deo 


Oi  r^  r^  CD 

^  ^  lO  o 

■*  CO  i-i  o 

rH  00(N 


CO  CO  CD  CO  o 
CO  00  CO  »o  o 

00  o  CO  ^  o 
1— I .— I  o  oco 


(M  ^  t^  C5  (M 

t^  ■*  O  CO  "tH 
lO  O  C^  T— I  CO 

T-H  O  O  O  <M 


r^  »o  r— I  C2  CO 

1—1  CO  CO  O  CD 
(M  t^  <M  T-H  o 

^  O  O  O  (M 


O  CO  CO  1— I  CO  o 

o  CO  00  T-H  o  o 

lO  00  lO  <M  i— I  O 

1-1  o  doo(M 


O  lO-*  (M  ■*  O 

o  t^  t^  t^  00  CO 

C^  CD  ^  >— I  O  CD 
1— I  O  O  O  O  '-H 


t^  t^  CO  iM  -^  O 
t— I  1— t  CD  CO  CD  lO 

a;  kO  CO  r-H  o  (M 

O  O  O  O  O  T-H 


O  O  CO  o  o  o  o 
O  O  CO  lO  O  lO  o 
O  lO  CO  (M  T-H  O  O 

^  o  o  o  o  o  ^ 


t-  CO  (M  t^  t^  CO  t^ 

CD  CO  (M  CD  CD  CO  CD 
CD  CO  C^  >— I  O  O  CD 

d>  d>  d  CD  d>  d>  d> 


s 

II 

a 

o  o  r^  vo  o  lO  o 

O  >0  CD  (M  lO  (M  O 
lO  (M  »-<  1-^  O  O  lO 

ooooooo 

--H  iM  CO  "*  O  O 


184 


REDUCING   TIME   IN   RELIABILITY   STUDIES 


185 


Table  III  —  Value  of  r  Required  to  Meet  the  Specification 
(a*,  P*)  FOR  Procedures  of  Type  R2  (g  =  0) 


a* 

p* 

1.05 

1.10 

1.15 

1.20 

1.25 

1.30 

1.35 

1.40 

0 

1.45 
0 

1.50 

2.00 

2.50 
0 

3.00 

0.50 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0.55 

14 

4 

2 

2 

1 

1 

1 

1 

1 

1 

1 

1 

1 

0.60 

55 

15 

7 

5 

3 

3 

2 

2 

2 

1 

1 

1 

1 

0.65 

126 

33 

16 

10 

7 

5 

4 

3 

3 

3 

1 

1 

1 

0.70 

232 

61 

29 

17 

12 

9 

7 

6 

5 

4 

2 

1 

1 

0.75 

383 

101 

47 

28 

19 

14 

11 

9 

7 

6 

3 

2 

1 

0.80 

596 

157 

73 

43 

29 

21 

17 

13 

11 

9 

4 

2 

2 

0.85 

903 

238 

111 

65 

44 

32 

25 

20 

16 

14 

5 

3 

3 

0.90 

1381 

363 

169 

100 

67 

49 

37 

30 

25 

21 

8 

5 

4 

0.95 

2274 

597 

278 

164 

110 

80 

61 

49 

40 

34 

12 

7 

5 

0.99 

4549 

1193 

556 

327 

219 

160 

122 

98 

80 

68 

24 

14 

10 

It  i.s  also  unnecessary  to  prepare  a  separate  table  for  the  average  ex- 
periment time  for  procedures  of  type  R2  since  for  g  =  0  the  exact  values 
can  be  obtained  by  substituting  the  appropriate  value  of  n  in  the  ex- 
pressions appearing  in  Table  II  in  the  row  marked  n  =  oo  .  For  example, 
for  /(  =  2,  /•  =  1  and  a  =  1  the  exact  value  for  ^  =  0  is  0.500  62/2  = 
0.250  62 ,  and  for  n  =  3,  r  =  4,  a  =  00  the  exact  value  for  g  =  0  is 
4.000  62/3  =  1.333  62 .  It  should  be  noted  that  for  procedures  of  type  R2 
we  need  not  restrict  our  attention  to  the  cases  r  ^  n  but  can  also  con- 
sider r  >  //. 

Table  III  shows  the  value  of  r  recjuired  to  meet  the  specilication 
(a*,  F*)  with  a  procedure  of  type  R2  for  various  selected  values  of  a* 
and  P*. 


procedures  of  type  R3  —  sequential,  replacement 

Let  D{t)  denote  the  absolute  difference  between  the  number  of  fail- 
ures produced  by  the  two  processes  at  any  time  t.  The  sequential  pro- 
cedure is  as  follows: 

"Stop  the  test  as  soon  as  the  inequality 


Dit)  ^ 


In  [P*/{1  -  P*)] 


In 


a 


(3) 


is  satisfied.  Then  select  the  population  with  the  smaller  number  of  fail- 
ures as  the  better  one." 

To  get  the  best  results  we  will  choose  (a*,  P*)  so  that  the  right  hand 
member  of  the  inequality  (3)  is  an  integer.  Otherwise  we  would  be  operat- 
ing with  a  higher  value  of  P*  (or  a  smaller  value  of  a*)  than  was  specified. 


186 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


Table   IV  —  Average   Experiment   Time   and   Probability   of   a 

Correct  Selection  —  Procedure  Type  R3 

(a*   =   2,  P*   =  0.800,  ^   =  0) 

(Multiply  each  average  time  entry  by  d^) 


n 

a  =  1 

a  =  2 

a  =   00 

1 

2.000 

2.400 

2.000 

2 

1.000 

1.200 

1.000 

3 

0.667 

0.800 

0.667 

4 

0.500 

0.600 

0.500 

10 

0.200 

0.240 

0.200 

20 

0.100 

0.120 

0.100 

oc 

2.000/w 

2.400/n 

2.000/n 

Probability 

0.500 

0.800 

1.000 

For  example,  we  might  choose  a*  =  2  and  P*  =  0.800.  For  procedures 
of  type  R3  the  probability  of  a  correct  selection  is  again  completely  in- 
dependent of  n;  here  it  depends  only  on  the  true  value  of  the  ratio  a. 
The  average  experiment  time  depends  strongly  on  n  and  only  to  a  limited 
extent  on  the  true  value  of  the  ratio  a.  Table  IV  gives  these  quantities 
for  a  =  1,  a  =  2,  and  a  =  00  for  the  particular  specification  a*  =  2, 
p*  =  0.800  and  for  the  particular  value  ^  =  0. 

efficiency 

We  are  now  in  a  position  to  compare  the  efficiency  of  two  different 
types  of  procedures  using  the  same  value  of  n.  The  efficiency  of  Ri  rela- 
tive to  R2  is  the  reciprocal  of  the  ratio  of  their  average  experiment  time. 
This  is  given  in  Table  V  for  a*  =  2,  P*  =  0.800,  r  =  4  and  n  =  4,  10,  20 
and  00 .  By  Table  I  the  value  P*  =  0.800  is  not  attained  for  n  <  4. 

In  comparing  the  sequential  and  the  nonsequential  procedures  it  was 
found  that  the  slight  excesses  in  the  last  column  of  Table  I  over  0.800 

Table  V  —  Efficiency  of  Type  Ri  Relative  to 

Type  R2 
{a*  =  2,  P*  =  0.800,  r  =  4:,g  =  0) 


{ 


n 

a  =   1 

a  =  2 

a  =   00 

4 
10 
20 

00 

0.501 
0.837 
0.925 
1.000 

0.495 
0.836 
0.917 
1.000 

0.480 
0.835 
0.922 
1.000 

I 


REDUCING  TIME   IN   RELIABILITY   STUDIES 


187 


Table  VI 

—  Efficiency  of 
(«*  =  2,  P* 

Adjusted  Ri  Relative  To  R^ 
=  0.800,  ^  =  0) 

n 

a  =  1 

a  =  2 

a  =    00 

4 
10 
20 

00 

0.615 
0.754 
0.818 
0.873 

0.575 
0.708 
0.768 
0.822 

0.419 
0.528 
0.573 
0.612 

had  an  effect  on  the  efficiency.  To  make  the  procedures  more  comparable 
the  values  for  r  =  3  and  r  =  4  in  Table  I  were  averaged  with  values  p 
and  1  —  p  computed  so  as  to  give  a  probability  of  exactly  0.800  at  a  =  a*. 
The  corresponding  values  for  the  average  experiment  time  were  then 
averaged  with  the  same  values  p  and  1  —  p.  The  nonsequential  pro- 
cedures so  altered  will  be  called  "adjusted  procedures."  The  efficiency 
of  the  adjusted  Ri  relative  to  Rz  is  given  in  Table  VI. 

In  Table  VI  the  last  row  gives  the  efficiency  of  the  adjusted  procedure 
7^2  relative  to  Rz  .  Thus  we  can  separate  out  the  advantage  due  to 
the  replacement  feature  and  the  advantage  due  to  the  sequential  fea- 
ture. Table  VII  gives  these  results  in  terms  of  percentage  reduction  of 
average  experiment  time. 

We  note  that  the  reduction  due  to  the  replacement  feature  alone  is 
greatest  for  small  n  and  essentially  constant  with  a  while  the  reduction 


Table  VII  —  Per  Cent  Reduction  in  Average  Experiment  Time 
DUE  TO  Statistical  Techniques 

(a*  =  2,P*  =  0.800,  ^  =  0) 


a 

K 

Reduction  due  to 

Replacement 

Feature  Alone 

Reduction  due  to 

Sequential 
Feature  Alone 

Reduction 

due  to  both 

Replacement 

and  Sequential 

Features 

1 

4 
10 
20 

00 

29.5 

13.7 

6.3 

0.0 

12.7 
12.7 
12.7 
12.7 

38.5 
24.6 
18.2 
12.7 

2 

4 
10 
20 

00 

30.1 

13.9 

6.6 

0.0 

17.8 
17.8 

17.8 
17.8 

42.5 
29.2 
23.2 
17.8 

cc 

4 
10 
20 

00 

31.5 

13.6 

6.3 

0.0 

38.8 
38.8 

38.8 
38.8 

58.1 
47.2 
42.7 
38.8 

188  THE   BELL   SYSTEM   TECHNICAL  JOURNAL,    JANUARY    1956 

due  to  the  sequential  feature  alone  is  greatest  for  large  a  and  is  inde- 
pendent of  n.  Hence  if  the  initial  sample  size  per  process  n  is  large  we 
can  disregard  the  replacement  techniciue.  On  the  other  hand  the  true 
value  of  a  is  not  known  and  hence  the  advantage  of  sequential  experi- 
mentation should  not  be  disregarded. 

The  formulas  used  to  compute  the  accompanying  tables  are  given  in 
Addendum  2. 

ACKNOWLEDGEMENT 

The  author  wishes  to  thank  Miss  Marilyn  J.  Huyett  for  considerable 
help  in  computing  the  tables  in  this  paper.  Thanks  are  also  due  to 
J.  W.  Tukey  and  other  staff  members  for  constructive  criticism  and 
numerical  errors  they  have  pointed  out. 

Addendum  1 

In  this  addendum  we  shall  consider  the  more  general  problem  of  select- 
ing the  best  of  k  exponential  populations  treated  on  a  higher  mathemati- 
cal level.  For  k  =  2  this  reduces  to  the  problem  discussed  above. 

DEFINITIONS   AND   ASSUMPTIONS 

There  are  given  k  populations  H,  (^  =  1,  2,  •  •  •  ,  k)  such  that  the  life- 
times of  units  taken  from  any  of  these  populations  are  independent 
chance  variables  with  the  exponential  density  (1)  with  a  common  (known 
or  unknown)  location  parameter  g  ^  0.  The  distributions  for  the  k  popu- 
lations are  identical  except  for  the  unknown  scale  parameter  6  >  0  which 
may  be  different  for  the  k  different  populations.  We  shall  consider  three 
different  cases  with  regard  to  g. 

Case  1 :  The  parameter  g  has  the  value  zero  (g  =  0). 

Case  2:  The  parameter  g  has  a  positive,  known  value  (g  >  0). 

Case  3:  The  parameter  g  is  unknown  (g  ^  0). 
Let  the  ordered  values  of  the  k  scale  parameters  be  denoted  by 

di^  e.-^  ■■■  ^  dk  (4) 

where  equal  values  may  be  regarded  as  ordered  in  any  arbitrary  manner. 
At  any  time  /  each  population  has  a  certain  number  of  failures  associated 
with  it.  Let  the  ordered  values  of  these  integers  be  denoted  by  ri  =  ri{t) 
so  that 


I 


ri  g  r2  ^  •  •  •  ^  r-fc  (5)  ^ 


i 


REDUCING   TIME   IN    RELIABILITY   STUDIES  189 

For  each  unit  the  life  beyond  its  guarantee  period  will  be  referred  to 
as  its  Poisson  life.  Let  Li{t)  denote  the  total  amount  of  Poisson  life 
observed  up  to  time  t  in  the  population  with  Vi  failures  (z  =  1,  2,  •  •  •  ,  fc). 
If  two  or  more  of  the  r^  are  equal,  say  Vi  =  rj+i  =  •  •  •  =  r^+y ,  then  we 
shall  assign  r,  and  L;  to  the  population  with  the  largest  Poisson  life, 
ri+i  and  L^+i  to  the  population  with  the  next  largest,  •  •  •  ,  ri+_,  and  Lj+,- 
to  the  population  with  the  smallest  Poisson  life.  If  there  are  two  or  more 
equal  pairs  (ri ,  Li)  then  these  should  be  ordered  by  a  random  device 
giving  equal  probability  to  each  ordering.  Then  the  subscripts  in  (5)  as 
well  as  those  in  (4)  are  in  one-to-one  correspondence  with  the  k  given 
populations.  It  should  be  noted  that  Li(t)  ^  0  for  all  i  and  any  time 
t  ^  0.  The  complete  set  of  quantities  Li{t)  {i  =  1,  2,  •  •  •  ,  k)  need  not 
be  ordered.  Let  a  =  61/62  so  that,  since  the  6i  are  ordered,  a  ^  1. 

We  shall  further  assume  that : 

1 .  The  initial  number  n  of  units  put  on  test  is  the  same  and  the  start- 
ing time  is  the  same  for  each  of  the  k  populations. 

2.  Each  replacement  is  assumed  to  be  a  new  unit  from  the  same  popu- 
lation as  the  failure  that  it  replaces. 

3.  Failures  are  assumed  to  be  clearly  recognizable  without  any  chance 
of  error. 

SPECIFICATIONS   FOR   CASE    1 :   gf    =    0 

Before  experimentation  starts  the  experimenter  is  asked  to  specify  two 
constants  a*  and  P*  such  that  a*  >  1  and  l'^  <  P*  <  1.  The  procedure 
Ri  =  Rsin),  which  is  defined  in  terms  of  the  specified  a*  and  P*,  has 
the  property  that  it  will  correctly  select  the  population  with  the  largest 
scale  parameter  with  probability  at  least  P*  whenever  a  ^  a*.  The  initial 
number  n  of  units  put  on  test  may  either  be  fixed  by  nonstatistical  con- 
siderations or  may  be  determined  by  placing  some  restriction  on  the 
average  experiment  time  function. 

Rule  Rs : 

"Continue  experimentation  with  replacement  until  the  inequality 

k 

^  ^*-(^.-a)  ^   (1  _  p*)/p*  (6) 

i=2 

is  satisfied.  Then  stop  and  select  the  population  with  the  smallest  num- 
ber of  failures  as  the  one  having  the  largest  scale  parameter." 


190 


THE    BELL    SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 


Remarks 

1.  Since  P*  >  Y2  then  (1  —  P*)/P*  <  1  and  hence  no  two  popula- 
tions can  have  the  same  vahie  ri  at  stopping  time. 

2.  For  A:  =  2  the  inequality  (6)  reduces  to  the  inequalitj^  (3). 

3.  The  procedure  7^3  terminates  onl}^  at  a  failure  time,  never  between 
failures,  since  the  left  member  of  (G)  depends  on  t  only  through  the 
quantities  7-i{t). 

4.  After  experimentation  is  completed  one  can  make,  at  the  lOOP  per 
cent  confidence  level,  the  confidence  statement 


ds  ^  di  S  a*  9,     (or     di/a"" 


^  ds  S  e,) 


(7) 


where  6s  is  the  scale  parameter  of  the  selected  population. 


Numerical  Illustrations 


»l/4 


Suppose  the  preassigned  constants  are  P*  =  0.95  and  a*  =  19' 
2.088  so  that  (1  -  P*)/P*  =  ^9-  Then  for  A;  =  2  the  procedure  is  to 
stop  when  r-i  —  ri  ^  4.  For  A;  =  3  it  is  easy  to  check  that  the  procedure 
reduces  to  the  simple  form:  "Stop  when  ?'2  —  ri  ^  5".  For  A;  >  3  either 
calculations  can  be  carried  out  as  experimentation  progresses  or  a  table 
of  stopping  values  can  be  constructed  before  experimentation  starts. 
For  A:  =  4  and  A;  =  5  see  Table  VIII. 

In  the  above  form  the  proposed  rule  is  to  stop  Avhen,  for  at  least  one 


Table  VIII  —  Sequential  Rule  for  P*  =  0.95,  a*  =   19 
A:  =  4  fc  =  5 


1/4 


r2  —  ri 

rs  —  ri 

n  —  ri 

5 

5 

9 

5 

6 

6 

6 

6 

6 

ri  —  ri 

ra  —  ri 

n  —  ci 

Ti  —  n 

5 

5 

9 

10 

5 

5 

10 

10 

5 

6 

6 

8 

5 

6 

7 

7 

5 

7 

7 

7 

6 

6 

6 

6 

*  Starred  rows  can  be  omitted  without  affecting  the  test  since  every  integer  in 
these  rows  is  at  least  as  great  as  the  corresponding  integer  in  the  previous  row. 
They  are  shown  here  to  ilhistrate  a  systematic  method  which  insures  that  all  the 
necessary  rows  are  included. 


REDUCING   TIME   IN    RELIABILITY   STUDIES  191 

row  (say  row  j)  in  the  table,  the  observed  row  vector  (r^  —  Vi , 
Ts  —  Ti  ,  ■  ■  ■  ,  Vk  —  z'l)  is  such  that  each  comyonent  is  at  least  as  large  as 
the  corresponding  component  of  row  j. 

Properties  of  Rs  for  k  =  2  and  g  =  0 

For  A-  =  2  and  ^  =  0  the  procedure  Rs  is  an  example  of  a  Sequential 
Probability  Ratio  test  as  defined  by  A.  Wald  in  his  book.^  The  Average 
Sample  Number  (ASN)  function  and  the  Operating  Characteristics  (OC) 
function  for  Rs  can  be  obtained  from  the  general  formulae  given  by 
Wald.  Both  of  these  functions  depend  on  di  and  0-2  only  through  their 
ratio  a.  In  our  problem  there  is  no  excess  over  the  boundary  and  hence 
Wald's  approximation  formulas  are  exact.  When  our  problem  is  put  into 
the  Wald  framework,  the  symmetry  of  our  problem  implies  equal  proba- 
bilities of  type  1  and  type  2  errors.  The  OC  function  takes  on  comple- 
mentary values  for  any  point  a  =  61/62  and  its  reciprocal  62/61  .  We  shall 
therefore  compute  it  only  for  a  ^  1  and  denote  it  by  P{a).  For  a  >  1 
the  quantity  P(a)  denotes  the  probability  of  a  correct  selection  for  the 
true  ratio  a. 

The  equation  determining  Wald's  h  function  is 


1  +  a         1  +  a 
for  which  the  non-zero  solution  in  h  is  easily  computed  to  be 

h{a)  =  }^  (9) 


In 


a 


Hence  we  obtain  from  Wald's  formula  (3:43)  in  Reference  5 


s 

a 


Pia)  =  -^^  (10) 

where  s  is  the  smallest  integer  greater  than  or  equal  to 

S  =  In  [PV(1  -  P*)]/ln  a*  (11) 

In  particular,  for  a  =  1"^,  a*  and  00  we  have 

Pi^^)  =  1/2,         ^(«*)  ^  P*,         P(^)  =  1  (12) 

^\'e  have  written  P(l"^)  above  for  lim  P{x)  as  x  -^  1  from  the  right.  The 
procedure  becomes  more  efficient  if  we  choose  P  and  a*  so  that  *S'  is  an 
integer.  Then  s  ^  S  and  P(a*)  =  P*. 

Letting  F  denote  the  total  number  of  observed  failures  required  to 


192  THE    BELL    SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

terminate  the  experiment  we  obtain  for  the  ASN  function 

and,  in  particular,  for  a  =  1,  oo 

E(F;  1)  =  s-     and     E{F;  oo)  =  s  (14) 

It  is  interesting  to  note  that  for  s  =  1  we  obtain 

E{F;  a)  =  1  for  all  a  ^  1      (15) 

and  that  this  result  is  exact  since  for  s  =   1  the  right-hand  member  S  \ 

of  (3)  is  at  most  one  and  hence  the  procedure  terminates  with  certainty  ' 

immediately  after  the  first  failure.  ' 

As  a  result  of  the  exponential  assumption,  the  assumption  of  replace-  ; 

ment  and  the  assumption  that  ^  =  0  it  follows  that  the  intervals  between  \ 

failures  are  independently  and  identically  distributed.  For  a  single  popu-  ' 

lation  the  time  interval  between  failures  is  an  exponential  chance  vari-  ; 

able.  Hence,  for  two  populations,  the  time  interval  is  the  minimum  of  j 

two  exponentials  which  is  again  exponential.   Letting  r  denote  the  i 

(chance)  duration  of  a  typical  interval  and  letting  T  denote  the  (chance)  j 
total  time  needed  to  terminate  the  procedure,  Ave  have 

E{T;  a,  62)  =  E{F;  a)E(r;  a,  d^)  =  E{F;  a)  (^^^  (f^)      (16) 

I 

Hence  Ave  obtain  from  (13)  and  (14) 

E{T;  a,  02)  =  -  -^  ^^^  for  a  >  1     (17) 

n  a  —  1  a*  +  1 

E{T;  1,  d,)  =  ^        and         E{T;  <^,  0,)  =  ^  (18> 

For  the  numerical  illustration  treated  above  Avith  k  =  2  we  have 

na)  =  ^-^  (19) : 

P(l+)  =  ^;         P(2.088)  =  0.95;         P(oo)  =  1  (20) 

EiF-a)  =  4^^4^  =  4^--+  Vy  +  '^  (21) 

a—   la*-f-l  a*-t-l 

E{F;  1)  =  16.0;         /iXF;  2.088)  =  10.2;         E{F;  00)  =  4  (22), 


REDUCING   TIME    IN    RELIABILITY   STUDIES  193 

E(T;  1,  ^2)  =  —  ;        E{T;  2.088,  6^  =  —  ; 

n  n  (23) 

n 

For  /.•  >  2  the  proposed  procedure  is  an  application  of  a  general  se- 
quential rule  for  selecting  the  best  of  A-  populations  which  is  treated  in 
[1].  Proof  that  the  probability  specification  is  met  and  bounds  on  the 
probability  of  a  correct  decision  can  be  found  there. 

CASE     2:    COMMON   KNOWN   ^    >    0 

In  order  to  obtain  the  properties  of  the  sec^uential  procedure  R:>.  for 
this  case  it  will  be  convenient  to  consider  other  sequential  procedures. 
Let  (S  =  1/6-2  —  1/^1  so  that,  since  the  di  are  ordered,  jS  ^  0.  Let  us 
assume  that  the  experimenter  can  specify  three  constants  a*,  /3*  and 
P*  such  that  a*  >  1,  /3*  >  0  and  ^  2  <  -P*  <  1  ai^d  a  procedure  is  de- 
sired which  will  select  the  population  with  the  largest  scale  parameter 
with  probability  at  least  P*  whenever  we  have  both 

a  ^  a*     and     i3  ^  /3* 

The  following  procedure  meets  this  specification. 

Rule  Rs': 

"Continue  experimentation  with  replacement  until  the  inec^uality 

fi  «*-(^i-'-i>e-^*(^i-^i)^  (l_p*)/p*  (24) 

1=2 

is  satisfied.  Then  stop  and  select  the  population  with  the  smallest  nimiber 
of  failures  as  the  one  having  the  largest  scale  parameter.  If,  at  stopping 
time,  two  or  more  populations  have  the  same  value  ri  then  select  that 
particular  one  of  these  with  the  largest  Poisson  life  Li  ." 

Remarks 

1 .  For  k  =  2  the  inequality  reduces  to 

(r,  -  n)  In  a*  +  (Li  -  L2)  13*  ^  In  [P*/a  -  P*)]  (25) 

If  <7  =  0  then  Li  =  Li  for  all  t  and  the  procedure  R/  reduces  to  R3  . 

2.  The  procedure  R/  may  terminate  not  only  at  failures  but  also  be- 
tween failures. 


194  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

3.  The  same  inequality  (24)  can  also  be  used  if  experimentation  is 
carried  on  without  replacement,  one  advantage  of  the  latter  being  that 
there  is  less  bookkeeping  involved.  In  this  case  there  is  a  possibility 
that  the  units  will  all  fail  before  the  inequality  is  satisfied  so  that  the 
procedure  is  not  yet  completely  defined  for  this  case.  One  possibility 
in  such  a  situation  is  to  continue  experimentation  with  new  units  from 
each  population  until  the  inequality  is  satisfied.  Such  a  procedure  will 
terminate  in  a  finite  time  with  probability  one,  i.e.,  Prob{  T  >  To}  -^0 
as  To  — >  00,  and  the  probability  specification  will  be  satisfied. 

4.  A  procedure  R3  (ni  ,  n-z ,  ■  •  •  ,  rik  ,  ti ,  t2 ,  •  •  •  ,  tk)  using  the  same 
inequality  (24)  but  based  on  dilTerent  initial  sample  sizes  and/or  on 
different  starting  times  for  the  initial  samples  also  satisfies  the  above 
probability  specification.  In  the  case  of  different  starting  times  it  is 
required  that  the  experimenter  wait  at  least  g  units  of  time  after  the  last 
initial  sample  is  put  on  test  before  reaching  any  decision. 

0.  One  disadvantage  of  R3  is  that  there  is  some  (however  remote) 
possibility  of  terminating  while  ri  =  r2  .  This  can  be  avoided  by  adding 
the  condition  r^  >  n  to  (24)  but,  of  course,  the  average  experiment  time 
is  increased.  Another  way  of  avoiding  this  is  to  use  the  procedure  R3 
which  depends  only  on  the  number  of  failures;  the  effect  of  using  R3 
when  g  >  0  will  be  considered  below. 

6.  The  terms  of  the  sum  in  (24)  represent  likelihood  ratios.  If  at  any 
time  each  term  is  less  than  unity  then  we  shall  regard  the  decision  to 
select  the  population  with  n  failures  and  Li  units  of  Poisson  life  as  opti- 
mal. Since  (1  —  P*)/P*  <  1  then  each  term  must  be  less  than  unity  at 
termination. 

Properties  of  Procedure  Rz  for  k  =  2  p 

The  OC  and  ASN  functions  for  Rs  will  be  approximated  by  comparing 
R3'  with  another  procedure  R/  defined  below.  We  shall  assume  that  P* 
is  close  to  unity  and  that  g  is  small  enough  (compared  to  d^)  so  that  the 
probability  of  obtaining  two  failures  within  g  imits  of  time  is  small 
enough  to  be  negligible.  Then  we  can  write  approximately  at  termination 

Li^nT  -  r,g        {i  =  1,  2,  •  •  •  ,  A:)  (26) 

and 

Li  -  Li  ^  (r,  -  r,)g  (i  =  2,  3,  •  •  •  ,  A:)     (27) 

Substituting  this  in  (24)  and  letting 

5*  =  a*  c^*"  (28) 

suggests  a  new  rule,  say  R/' ,  which  we  now  define. 


REDUCING   TIME    IN    RELIABILITY    STUDIES  195 

h'ule  R/ 

"Continue  experimentation  with  replacement  until  the  inequality 

k 

X  6*-(^i-'-i)  ^  (1  -  P*)/P*  (29) 


is  satisfied.  Then  stop  and  select  the  population  with  n  failures  as  the 
one  with  the  largest  scale  parameter." 

For  rule  Rz"  the  experimenter  need  only  specify  P*  and  the  smallest 
value  5*  of  the  single  parameter 

8  =  ^'  e''''"''-''"'''  =  ae'^  (30) 

62 

that  he  desires  to  detect  with  probability  at  least  P*. 

We  shall  approximate  the  OC  and  ASX  function  of  R/'  for  k  =  2 
by  computing  them  under  the  assumption  that  (27)  holds  at  termina- 
tion. The  results  will  be  considered  as  an  approximation  for  the  OC  and 
ASN  functions  respectively  of  R/  for  /,■  =  2.  The  similarity  of  (29) 
and  (6)  immediately  suggests  that  we  might  replace  a*  by  5*  and  a  by 
5  in  the  formulae  for  (6).  To  use  the  resulting  expressions  for  R^  we 
would  compute  5*  as  a  function  of  a*  and  /3*  by  (28)  and  5  as  a  function 
of  a  and /3  by  (30). 

The  similarity  of  (29)  and  (6)  shows  that  Z„  (defined  in  Reference  5, 
page  170)  under  (27)  with  gr  >  0  is  the  same  function  of  5*  and  5  as  it 
is  of  a*  and  a  when  g  =  0.  To  complete  the  justification  of  the  above 
result  it  is  sufficient  to  show  that  the  individual  increment  ^  of  Z„  is  the 
same  function  of  5*  and  8  under  (27)  with  ^  >  0  as  it  is  of  a*  and  a 
when  ^  =  0.  To  keep  the  increments  independent  it  is  necessary  to  as- 
sociate each  failure  with  the  Poisson  life  that  follows  rather  than  with 
the  Poisson  life  that  precedes  the  failure.  Neglecting  the  probability 
that  any  two  failures  occur  ^^•ithin  g  units  of  time  we  have  two  values  for 
z,  namely 

^      -(.nt-g)/ei    -ntl$2 

z  =  log^^^ =  -log  5  (31) 

and,  interchanging  61  and  ^2 ,  gives  z  —  log  5.  Moreover 


196  THE    BELL    SYSTEM    TECHNICAL    JOURNAL,    JANUARY    1956 

r  r  -  e-(— «)/«v"^^^^  dx  dy 

Jg  Jg  6-1 


Prob  \z  =   -logSj 


^2     -0[92(n-l)+9l"l/9lfl2    _i_   ^1     -H9in+Bi(n-l)]l9ie2,) 

-  e  +  -  e  /o9\ 


1  +  5 


Thus  the  OC  and  ASN  functions  under  (27)  with  g  >  0  bear  the  same 
relation  to  5*  and  5  as  they  do  to  a*  and  a  when  ^  =  0.  Hence,  letting 
w  denote  the  smallest  integer  greater  than  or  equal  to 

^  In  [P*/(l  -  P*)]  ^  \n[P*/{l-P*)] 

In  8*  gl3*  +  In  a*  ^'  ' 

we  can  write  (omitting  P*  in  the  rule  description)  | 

7^15;  /?/  («*,  /5*){  ^  P{5;  /^.^"(S*)!  ^  ^-^^^  (34) 

<w. I   ■ — -  tor  5  >  1      {So) 

^    \8  -  l/\5"'  +1/ 


w~  for  5=1 

W'e  can  approximate  the  average  time  between  failures  by 


I 


and  the  average  experiment  time  by  « 

E{T;  /?/(«*,  ^*)}  ^  E{F;  R,'(a*,  0*)\  [^.^ f^'^ _^ ^'^,        (37) 

n{Oi  -T  02  -f-  zg) 

Since  5  ^  1  then  5"(1  +  5")  is  an  increasing  function  of  w  and  by 
(33)  it  is  a  non-increashig  function  of  5*.  By  (28)  5*  ^  a*  and  hence, 
if  we  disregard  the  approximation  (34), 

P{8;  AV(«*)1  -  ^!^{py^/_p.^y..n^*  ^  P{S;R/m}    (38) 

Clearly  the  rules  Ri{a*,  P*)  and  R/  {a*,  P*)  are  equivalent  so  that 
for  g  >  0  we  haA-e 

P{8;R-s{a*)}   ^  P{8;R/ia*)]  (39) 


REDUCING   TIME    IN   RELIABILITY   STUDIES  197 

and  hence,  in  particular,  letting  8  =  8*  in  (38)  we  have 

P{8*;R,(a*)}  ^  P{8*;R,"(8*)]  ^  P*  (40) 

since  the  right  member  of  (34)  reduces  to  P*  when  W  is  an  integer  and 

5  =  5*.  The  error  in  the  approximations  above  can  be  disregarded  when 
g  is  small  compared  to  02  .  Thus  we  have  shown  that  for  small  values  of 
g/d2  the  probability  specification  based  on  (a*,  ^*,  P*)  is  satisfied  in  the 
sense  of  (40)  if  we  use  the  procedure  Rsia*,  P*),  i.e.,  if  we  proceed  as  if 

It  would  be  desirable  to  show  that  w^e  can  proceed  as  if  g  =  0  for  all 
values  of  g  and  P*.  It  can  be  shown  that  for  swfficiently  large  n  the  rule 
Ri{a*,  P*)  meets  it  specification  for  all  g.  One  effect  of  increasing  n 
is  to  decrease  the  average  time  E{t)  between  failures  and  to  approach 
the  corresponding  problem  without  replaceme^it  since  g/E{T)  becomes 
large.  Hence  we  need  only  show  that  Ri{a*,  P*)  meets  its  specification 
for  the  corresponding  problem  without  replacement.  If  we  disregard  the 
information  furnished  by  Poisson  life  and  rely  solely  on  the  counting  of 
failures  then  the  problem  reduces  to  testing  in  a  single  binomial  whether 

6  =  di  for  population  IIi  and  6  =  do  for  population  112  or  vice  versa.  Let- 
ting p  denote  the  probability  that  the  next  failure  arises  from  111  then 
we  have  formally 

tia'-V  =  -. — ; —  versus  Hi-.p  = 


1  +  a  ^        1  +  a 

For  preassigned  constants  a*  >  I  and  P*  (V2  <  P*  <  1)  the  appropri- 
ate sequential  likelihood  test  to  meet  the  specification: 

"Probability  of  a  Correct  Selection  ^  P*  whenever  a  ^  a*"  (41) 
then  turns  out  to  be  precisely  the  procedure  Rsia*,  P*).  Hence  we  may 
proceed  as  if  gr  =  0  when  n  is  sufficiently  large. 

The  specifications  of  the  problem  may  be  given  in  a  different  form. 
Suppose  01*  >  02*  are  specified  and  it  is  desired  to  haxe  a  probability  of  a 
correct  selection  of  at  least  P*  whenever  ^1  ^  0i*  >  02*  ^  02  .  Then  we 
can  form  the  following  sequential  likelihood  procedure  R3*  which  is 
more  efficient  than  Rsia*,  P*). 

Rule  /?3*.- 

"Continue  experimentation  without  replacement  until  a  time  t  is 
reached  at  which  the  inequality 


198  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

is  satisfied.  Then  stop  and  select  the  population  with  ri  failures  as  the 
population  with  d  =  di". 

It  can  be  easily  shown  that  the  greatest  lower  bound  of  the  bracketed 
quantity  in  (42)  is  0i*/^2*.  Hence  for  di*/d2*  =  a*  and  P*  >  i  2  the  time 
required  by  Rz*{6i*,  62*,  P*)  ivill  always  be  less  than  the  time  required 
by  R,(a*,P*). 

Another  type  of  problem  is  one  in  which  we  are  given  that  6  =  di* 
for  one  population  and  d  =  62*  for  the  A;  —  1  others  where  6]*  >  62*  are 
specified.  The  problem  is  to  select  the  population  with  6  =  di*.  Then 
(42)  can  again  be  used.  In  this  case  the  parameter  space  is  discrete  with 
k  points  only  one  of  which  is  correct.  If  Rule  R3*  is  used  then  the 
probability  of  selecting  the  correct  point  is  at  least  P*. 

Equilibrium  Approach  When  Failures  Are  Replaced 

9 

Consider  first  the  case  in  which  all  items  on  test  are  from  the  same 

exponential  population  with  parameters  (6,  g).  Let  Tnj  denote  the  length 
of  the  time  interval  between  the  j^^  and  the  j  +  1^*  failures,  (j  =  0, 
1,  •  •  •  ),  where  n  is  the  number  of  items  on  test  and  the  0*''  failure  de- 
notes the  starting  time.  As  time  increases  to  infinity  the  expected  number 
of  failures  per  unit  time  clearly  approaches  n/(0  +  g)  which  is  called  the 
equilibrium  failure  rate.  The  inverse  of  this  is  the  expected  time  between 
failures  at  equilibrium,  say  E{Tn^).  The  question  as  to  how  the  quanti- 
ties E{Tnj)  approach  E(Tn^)  is  of  considerable  interest  in  its  own  right. 
The  following  results  hold  for  any  fixed  integer  71  ^  1  unless  explicitly 
stated  otherwise.  It  is  easy  to  see  that 

^^(^i)  ^  E{TnJ  ^  E(T„o)  (43) 

since  the  exact  values  are  respectively 

e       /,        e-^-^'^/^X  ^  g+d  ^        ,    d 


< 


^  9+  -  (44) 


n  —  1  \  n      /  n  n 

In  fact,  since  all  units  are  new  at  starting  time  and  since  at  the  time  of 
the  first  failure  all  units  (except  the  replacement)  have  passed  their 
guarantee  period  with  probability  one  then 

^(^i)  ^  E(Tnj)  S  E{Tn,)  (j  ^  0)     (45) 

If  we  compare  the  case  g  >  0  with  the  special  case  g  =  0  we  obtain 

E{2\j)  ^  -  (y=  1,2,  •••)     (46) 

n 


REDUCING   TIME   IN   RELIABILITY   STUDIES  199 

and  if  we  compare  it  with  the  non-replacement  case  {g/Q  is  large)  we 
obtain 

^(n,)  ^  -^.  (i  =  1,  2,  • .  •  ,  n  -  1).     (47) 

These  comparisons  show  that  the  difference  in  (46)  is  small  when  g/0  is 
small  and  for  j  <  n  the  difference  in  (47)  is  small  when  g/d  is  large. 

It  is  possible  to  compute  E{Tnj)  exactly  for  g  ^  0  but  the  computa- 
tion is  extremely  tedious  for  j  ^  2.  The  results  for  j  =  1  and  0  are  given 
in  (44).  Fori  =  2 


E(Tn2)    = 


n 


(n    +    2)(/i    -    1)      -(n-2)gie 


1  -  '    '    ': -e 

n 


+  Vl^iI  g-(«-i)p/^ ri-2_    -un-i),ie  I  {n>2) 


n  —  \  v?{n  —  1) 

and 


2{n-l)glB 


(48) 


E{T,.^  =  ^  -  ^  [1  -  ^e-'"  +  e-'"'\  (49) 

For  the  case  of  two  populations  with  a  common  guarantee  period  g 
we  can  write  similar  inequalities.  We  shall  use  different  symbols  a,  h  for 
the  initial  sample  size  from  the  populations  with  scale  parameters  Oi  ,  O2 
respectively  even  though  our  principal  interest  is  in  the  case  a  =  b  =  n 
say.  Let  Ta,b.j  denote  the  interval  between  the  j^^  and  j  -f  P*  fail- 
ures in  this  case  and  let  X,  =  l/di  (i  =  1,2).  We  then  have  for  all  values 
of  a  and  b 

[aXi  +  b\o]-'  ^  E(TaXj)  ^  E(Taxo) 

=  g  +  [aXi  +  b\,]-'     (j  =  0,1,2,  ■■■,  ^)     (50) 

J?(T  ^  (gl    +    g){e2    +    g)  .riN 

a{92  -h  9)  +  b{di  +  g) 

The  result  for  E(Ta,b.i)  corresponding  to  that  in  (43)  does  not  hold  if 
the  ratio  di/62  is  too  large;  in  particular  it  can  be  shown  that 

-0[(a-l)Xi+6X2l-l 


E{T.,b..)  =  ^       "^^       ^'  ^ 


aXi  +  6X2/ \(a  —  l)Xi  4-  6X2 


_  Xie 


aXi  +  6X2 


+  /       ^X2       Y  1  \r         x^e-''^'^^''-''''-'- 


(52) 


,aXi  -\-  bX2/\aXi  +  (&  —  1)^2  L  0X1  +  ^^2 

is  larger  than  E{Ta,h.J  for  a  =  6  =  1  when  ^/^i  =  0.01  and  g/di  =  0.10 


200  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 

SO  that  QilQi  =  10.  The  expression  (52)  reduces  to  that  in  (44)  if  we  set 
di  =  02  =  6  and  replace  a  and  h  by  n/2  in  the  resulting  expression. 

Corresponding  exact  expressions  for  E(Ta.b,j)  for  j  >  1  are  extremely 
tedious  to  derive  and  unwieldy  although  the  integrations  involved  are 
elementary.  If  we  let  g  —^  oo  then  we  obtain  expressions  for  the  non- 
replacement  case  which  are  relatively  simple.  They  are  best  expressed 
as  a  recursion  formula. 


E(.Ta,bj)    =    — ,      ,.       ETa-\,b,}-l 


+  m^  ^"—     ^^  =  '^ 


(53) 


EiT.,b.d  =         "^^  ^ 


aXi  +  6X2  (a  —  l)Xi  +  6X2 

I        0X2  1  (    h  >  ^^ 

"^  aXi  +  6X2  aXi  +  (6  -  1)X2  '     = 


(54) 


E(Tafij)  ^  g  +  di/a  fori  ^  a  and  j  =  0     (55) 

E{Ta,oJ  =  dr/(a  -j)  for  1  ^  i  ^  a  -  1     (56) 

Results  similar  to  (55)  and  (56)  hold  for  the  case  a  =  0.  The  above 
results  for  gr  =  00  provide  useful  approximations  for  E{Ta,b,j)  when  g 
is  large.  Upper  bounds  are  given  by  M 

E{Ta,bj)  ^  [aXi  +  (6  -  i)X2r  (i  =  1,  2,  •  •  •  ,  h)     (57) 

E(Ta.bj+b)  ^  [(a  -  j)Xr'  (i  =  1,  2,  •  .  •  ,  a  -  1).      (58) 

Duration  of  the  Experiment 

For  the  sequential  rule  R^'  with  k  =  2  we  can  now  write  down  approxi- 
mations as  well  as  upper  and  lower  bounds  to  the  expected  duration 
E{T)  of  the  experiment.  From  (50) 


I 


g  +  ..5^;^.\  s  E(T)  =  E  /?(r.,,) 


c-l 

n(Xi  -f  X2)  ^  '''^  '  ~  §  '^^^  "'"'^^  (59) 

+  \FA¥;  5)  -  c]i!;(T„,„,.) 


where  c  is  the  largest  integer  less  than  or  equal  to  E{F\  5).  The  right  ex- 
pression of  (59)  can  be  approximated  by  (53)  and  (54)  if  g  is  large.  If 
c  <  2n  then  the  upper  bounds  are  given  by  (57)  and  (58).  A  simpler 


j 


REDUCING   TIME   IN   RELIABILITY   STUDIES  201 

upper  bound,  which  holds  for  all  \'aliies  of  c  is  given  by 

E{T)  ^  E{F-  b)E{Tn,n..)  =  E{F;  8)  (g  +  ^^  (60) 

CASE   3:   COMMON    UNKNOWN   LOCATION   PARAMETER    ^    ^    0 

In  this  case  the  more  conservative  procedure  is  to  proceed  under  the 
assumption  that  </  =  0.  By  the  discussion  above  the  probability  require- 
ment will  in  most  problems  be  satisfied  for  all  ^  ^  0.  The  OC  and  ASN 
functions,  which  are  now  functions  of  the  true  value  of  g,  were  already 
obtained  above.  Of  course,  we  need  not  consider  values  of  g  greater  than 
the  smallest  observed  lifetime  of  all  units  tested  to  failure. 

Addendum  2 

For  completeness  it  would  be  appropriate  to  state  explicitly  some  of 
the  formulas  used  in  computing  the  tables  in  the  early  part  of  the  paper. 
For  the  nonsequential,  nonreplacement  rule  Ri  with  /c  =  2  the  proba- 
bility of  a  correct  selection  is 

P(a;  R,)  =    [    [   Mu,  OAfrix,  6,)  dy  dx  (61) 

where 

fXx,  e)  =  '-  C(l  -  e^'"y-'  e-^^"-^+^"^  (r  ^  n)     (62) 

and  C"  is  the  usual  combinatorial  symbol.  This  can  also  be  expressed  in 
the  form 

P{a;  R,)  =  1   -   (rC:r  Z     ^~^^"' 

;=i  n  -  r  -\-j  (63) 

C'-l{B[r,  n-r+l+a(n-r+  j)]}-' 

where  B[x,  y]  is  the  complete  Beta  function.  Eciuation  (66)  holds  for 
any  g  ^  0. 

For  the  rule  Ri  the  expected  duration  of  the  experiment  for  k  =  2 
is  given  by 

E{T)  =    r  x{fr(x,  d,)[l  -  Frix,  62)]  +  frix,  d,)[l  -  Fr(x,  ^i)] }  dx     (64) 

•'0 

where  frix,  6)  is  the  density  in  (62)  and  Fr{x,  B)  is  its  c.d.f.  This  can 


202  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

also  be  expressed  in  the  form 

^iKC^ZZt (-1)   c.-. c, .     

plus  another  similar  expression  in  which  6i  ,  a  are  replaced  by  62 ,  a~^ 
respectively.  For  ^  >  0  we  need  only  add  g  to  this  result.  This  result 
was  used  to  compute  E(T)  in  table  lA  f or  a  =  1  and  a  =  2.  For  a  =  oo 
the  expression  simplifies  to 

E{T)  =  e^rC:  ±  erl       ^~^^'^\  (66) 

which  can  be  shoAvn  to  be  equivalent  to 

E{T)  =  e,f: ^—  (67) 

REFERENCES 

1.  Bechhofer,  R.  E.,  Kiefer,  J.  and  Sobel,  M.,  On  a  Type  of  Sequential  Multiple 

Decision  Procedures  for  Certain  Ranking  and  Identification  Problems  with 
k  Populations.  To  be  published. 

2.  Birnbaum,  A.,   Statistical  methods  for  Poisson  processes  and  exponential 

populations,  J.  Am.  Stat.  Assoc,  49,  pp.  254-266,  1954. 

3.  Birnbaum,  A.,  Some  procedures  for  comparing  Poisson  processes  or  popula- 

tions, Biometrika,  40,  pp.  447-49, 1953. 

4.  Girshick,  M.  A.,  Contributions  to  the  theory  of  sequential  analj'sis  I,  Annals 

Math.  Stat.,  17,  pp.  123-43,  1946. 

5.  Wald,  A.,  Sequential  Analysis,  John  Wiley  and  Sons,  New  York,  1947. 


I 


A  Class  of  Binary  Signaling  Alphabets 

By  DAVID  SLEPIAN 

(Manuscript  received  September  27,  1955) 

A  class  of  binary  signaling  alphabets  called  "group  alphabets"  is  de- 
scribed. The  alphabets  are  generalizations  of  Hamming^ s  error  correcting 
codes  and  possess  the  following  special  features:  {1)  all  letters  are  treated 
alike  in  transmission;  {2)  the  encoding  is  simple  to  instrument;  (3)  maxi- 
mum likelihood  detection  is  relatively  simple  to  instrument;  and  (4)  in 
certain  practical  cases  there  exist  no  better  alphabets.  A  compilation  is  given 
of  group  alphabets  of  length  equal  to  or  less  than  10  binary  digits. 

INTRODUCTION 

This  paper  is  concerned  with  a  class  of  signahng  alphabets,  called 
"group  alphabets,"  for  use  on  the  symmetric  binary  channel.  The  class 
in  question  is  sufficiently  broad  to  include  the  error  correcting  codes  of 
Hamming,^  the  Reed-Muller  codes,"  and  all  "systematic  codes''.^  On 
the  other  hand,  because  they  constitute  a  rather  small  subclass  of  the 
class  of  all  binary  alphabets,  group  alphabets  possess  many  important 
special  features  of  practical  interest. 

In  particular,  (1)  all  letters  of  the  alphabets  are  treated  alike  under 
transmission;  (2)  the  encoding  scheme  is  particularly  simple  to  instru- 
ment; (3)  the  decoder  —  a  maximum  likelihood  detector —  is  the  best 
I  possible  theoretically  and  is  relatively  easy  to  instrument;  and  (4)  in 
certain  cases  of  practical  interest  the  alphabets  are  the  best  possible 
theoretically. 

It  has  very  recently  been  proved  by  Peter  Elias^  that  there  exist  group 
alphabets  which  signal  at  a  rate  arbitarily  close  to  the  capacity,  C,  of 
the  symmetric  binary  channel  with  an  arbitrarily  small  probability  of 
error.  Elias'  demonstration  is  an  existence  proof  in  that  it  does  not 
show  explicitly  how  to  construct  a  group  alphabet  signaling  at  a  rate 
greater  than  C  —  e  with  a  probability  of  error  less  than  5  for  arbitrary 
positive  5  and  e.  Unfortunately,  in  this  respect  and  in  many  others,  our 
understanding  of  group  alphabets  is  still  fragmentary. 

In  Part  I,  group  alphabets  are  defined  along  with  some  related  con- 

203 


204  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

cepts  necessary  for  their  understanding.  The  main  results  obtained  up 
to  the  present  time  are  stated  without  proof.  Examples  of  these  concepts 
are  given  and  a  compilation  of  the  best  group  alphabets  of  small  size 
is  presented  and  explained.  This  section  is  intended  for  the  casual  reader. 

In  Part  II,  proofs  of  the  statements  of  Part  I  are  given  along  with 
such  theory  as  is  needed  for  these  proofs. 

The  reader  is  assumed  to  be  familiar  with  the  paper  of  Hamming, 
the  basic  papers  of  Shannon*  and  the  most  elementary  notions  of  the 
theory  of  finite  groups. 

Part  I  —  Group  Alphabets  and  Their  Properties 

1.1    INTRODUCTION 

We  shall  be  concerned  in  all  that  follows  with  communication  over  the 
symmetric  binary  channel  shown  on  Fig.  1.  The  channel  can  accept 
either  of  the  two  symbols  0  or  1 .  A  transmitted  0  is  received  as  a  0  with 
probability  q  and  is  received  as  a  1  w'ith  probability  p  —  1  —  g :  a  trans- 
mitted 1  is  received  as  a  1  with  probability  q  and  is  received  as  a  0  with 
probability  p.  We  assume  0  ^  p  ^  ^^.  The  "noise"  on  the  channel 
operates  independently  on  each  symbol  presented  for  transmission.  The 
capacity  of  this  channel  is 

C  =  1  +  P  log2P  +  q  log29  bits/symbol  (1) 

By  a  K-leUer,  n-place  binary  signaling  alphabet  we  shall  mean  a  collec- 
tion of  K  distinct  sequences  of  n  binary  digits.  An  individual  sequence 
of  the  collection  will  be  referred  to  as  a  letter  of  the  alphabet.  The  integer 
K  is  called  the  size  of  the  alphabet.  A  letter  is  transmitted  over  the 
channel  by  presenting  in  order  to  the  channel  input  the  sequence  of  n 
zeros  and  ones  that  comprise  the  letter.  A  detection  scheme  or  detector  for 


INPUT  X  OUTPUT 


Fig.  1  —  The  symmetric  binary  channel. 


A   CLASS   OF   BINARY   SIGNALING   ALPHABETS 


205 


a  given  /v-letter,  n-place  alphabet  is  a  procedure  for  producing  a  sequence 
of  letters  of  the  alphabet  from  the  channel  output. 

Throughout  this  paper  we  shall  assume  that  signaling  is  accomplished 
with  a  given  /i-letter,  n-place  alphabet  by  choosing  the  letters  of  the 
alphabet  for  transmission  independently  with  equal  probability   l/K. 

Shannon^  has  shown  that  for  sufficiently  large  n,  there  exist  K-letter, 
n-place  alphabets  and  detection  schemes  that  signal  over  the  symmetric 
binary  chaimel  at  a  rate  R  >  C  —  e  for  arbitrary  £  >  0  and  such  that 
the  probability  of  error  in  the  letters  of  the  detector  output  is  less  than 
any  5  >  0.  Here  C  is  given  by  (1)  and  is  shown  as  a  function  of  p  in 
Fig.  2.  No  algorithm  is  known  (other  than  exhaustvie  procedures)  for 
the  construction  of  A'-letter,  /i-place  alphabets  satisfying  the  above 
inequalities  for  arbitrary  positive  8  and  e  except  in  the  trivial  cases  C  —  0 
and  C  =  1. 

1.2   THE   GROUP   -S„ 

There  are  a  totality  of  2"  different  w-place  binary  sequences.  It  is  fre- 
quently convenient  to  consider  these  sequences  as  the  vertices  of  a  cube 
of  unit  edge  in  a  Euclidean  space  of  n-dimensions.  For  example  the  5- 
place  sequence  0,  1,  0,  0,  1  is  associated  with  the  point  in  5-space  whose 


o.e 


0.6 


0.4 


0.2 


Fig.  2  —  The  capacity  of  the  symmetric  binary  channel. 
C  =  1  +  p  log2  p  +  {I  -  p)  log2  (1  -  p) 


206 


THE    BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 


coordinates  are  (0,  1,  0,  0,  1).  For  convenience  of  notation  we  shall  gen- 
erally omit  commas  in  writing  a  sequence.  The  above  5-place  sequence 
will  be  written,  for  example,  01001. 

We  define  the  product  of  two  n-ylace  hinarij  sequences,  aicii  •  •  •  a„  and 
^1^2  •  ■  •  bn  as  the  n-place  binary  sequence 

fli  +  hi  ,         a-i  ■]-  h-i  ,  ■  ■  •  ,  ttn  +  hn 

Here  the  a's  and  6's  are  zero  or  one  and  the  +  sign  means  addition 
modulo  2.  (That  is  0  +  0=1  +  1  =  0,  0+1  =  1+0=1) 
For  example,  (01101)  (00111)  =  01010.  With  this  rule  of  multiplication 
the  2"  w-place  binary  sequences  form  an  Abelian  group  of  order  2". 
The  elements  of  the  group,  denoted  by  Ti  ,  T'2 ,  •  •  •  ,  Tin,  say,  are  the 
n-place  binary  sequences ;  the  identity  element  I  is  the  sequence  000  •  •  •  0 
and 

IT,  =  Til  =  T.  ■        T,Tj  =  TjTr,        TiiTjT,)  =  iTiTj)Tk  ; 

the  product  of  any  number  of  elements  is  again  an  element;  every  ele- 
ment is  its  own  reciprocal,  Ti  =  Tf^,  TI  =  /.  We  denote  this  group 
by  Bn  . 

All  subgroups  of  Bn  are  of  order  2   where  k  is  an  integer  from  the  set 
0,  1,  2,  •  •  •  ,  n.  There  are  exactly 


N{n,  k)  = 


(2"  -  2")  (2"  -  2')  (2"  -  2')  •  •  •  (2"  -  2'-') 


(2^  -  2»)(2'^  -  20(2*  -  22) 
=  N(n,  n  —  k) 


{2"  -  2'-') 


(2) 


distinct  subgroups  of  Bn  of  order  2  .  Some  values  of  N(n,  k)  are  given  in 
Table  I. 


Table  I  — Some  Values  of  A^(n,  k),  the  Number  of  Subgroups 
OF  Bn  OF  Order  2''.  N(n,  k)   =  N{n,  n  —  k) 


n\k 

0 

1 

2 

3 

4 

5 

2 

3 

1 

3 

7 

7 

1 

4 

15 

35 

15 

1 

5 

31 

155 

155 

31 

1 

6 

63 

651 

1395 

651 

63 

7 

127 

2667 

IISU 

11811 

2667 

8 

255 

10795 

97155 

200787 

97155 

9 

511 

43435 

788035 

3309747 

3309747 

10 

1023 

174251 

6347715 

53743987 

109221651 

000 

000 

000 

000 

000 

000 

000 

100 

100 

100 

010 

010 

001 

no 

010 

001 

oil 

001 

101 

no 

on 

110 

101 

111 

on 

111 

111 

101 

A   CLASS   OF   BINARY   SIGNALING   ALPHABETS  207 

1.3   GROUP   ALPHABETS 

An  ?i-place  group  alphabet  is  a  7v-letter,  n-place  binary  signaling  alpha- 
bet whose  letters  form  a  subgroup  of  Bn  .  Of  necessity  the  size  of  an 
n-place  group  alphabet  is  /v  =  2  where  k  is  an  integer  satisfying  0  ^ 
k  ^  n.  By  an  (n,  k)-alphahet  we  shall  mean  an  n-place  group  alphabet  of 
size  2^.  Example:  the  N{3,  2)  =  7  distinct  (3,  2)-alphabets  are  given  by 
the  seven  columns 

(i)  (ii)  (iii)  (iv)  (v)  (vi)  (vii) 


(3) 


1.4      STANDARD   ARRAYS 

Let  the  letters  of  a  specific  (n,  /i:)-alphabet  be  Ai  =  /  =  00  •  •  •  0, 
Ao  ,  As  ,  •  ■  ■  ,  A^  ,  where  ju  =  2  .  The  group  Bn  can  be  developed  accord- 
ing to  this  subgroup  and  its  cosets: 

/,  A2,  A3,      ■■•  ,A^ 

S2 ,        S2A2 ,        S2A3 ,  •  •  •  ,  S2A^ 
Sz ,         S3A2 ,        S3A3 ,  •  •  •  ,  SsA^ 

Bn    =        ;  (4) 

Sr  f        SyA2 ,        SpAz ,  • '  •  ,  SfAfi 

In  this  array  every  element  of  Bn  appears  once  and  only  once.  The  col- 
lection of  elements  in  any  row  of  this  array  is  called  a  coset  of  the  (n,  k)- 
alphabet.  Here  *S2  is  any  element  of  B„  not  in  the  first  row  of  the  array, 
S3  is  any  element  of  Bn  not  in  the  first  two  rows  of  the  array,  etc.  The 
elements  S2 ,  S3 ,  •  •  •  ,  Sy  appearing  under  I  in  such  an  array  will  be 
called  the  coset  leaders. 

If  a  coset  leader  is  replaced  by  any  element  in  the  coset,  the  same  coset 
will  result.  That  is  to  say  the  two  collections  of  elements 

Si ,         ^1^2 ,         SiSz ;  ■  •  ■  ,  SiA^ 

and 

SiA,,  ,        (SiAu)A2 ,        (SiAMs  ,■■■  {SiAk)A, 

are  the  same. 


208  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    195G 

We  define  the  weight  Wi  =  w{Ti)  of  an  element,  Ti ,  of  Bn  to  be  the 
number  of  ones  in  the  n-place  binary  sequence  T,- . 

Henceforth,  unless  otherwise  stated,  we  agree  in  dealing  with  an  ar- 
ray such  as  (4)  to  adopt  the  following  convention: 

the  leader  of  each  coset  shall  be  taken  to  be  an  .  . 

element  of  minimal  weight  in  that  coset. 

Such  a  table  will  be  called  a  standard  array. 

Example:  Bi  can  be  developed  according  to  the  (4,  2)-alphabet  0000, 
1100,  0011,  nil  as  follows 


(6) 


0000 

1100 

0011 

nil 

1010 

Olio 

1001 

0101 

1110 

0010 

1101 

0001 

1000 

0100 

1011 

0111 

)W"ever, 

^^-e  should 

write. 

for  exan 

0000 

1100 

0011 

nil 

1010 

0110 

1001 

0101 

0010 

1110 

0001 

1101 

1000 

0100 

1011 

0111 

(7) 


The  coset  leader  of  the  second  coset  of  (6)  can  be  taken  as  any  element 
of  that  row  since  all  are  of  weight  2.  The  leader  of  the  third  coset,  how- 
ever, should  be  either  0010  or  0001  since  these  are  of  weight  one.  The 
leader  of  the  fourth  coset  should  be  either  1000  or  0100. 

1.5  THE   DETECTION    SCHEME 

Consider  now  communicating  with  an  (n,  fc) -alphabet  over  the  sym- 
metric binary  channel.  When  any  letter,  say  A,,  of  the  alphabet  is 
transmitted,  the  received  sequence  can  be  of  any  element  of  B„  .  We 
agree  to  use  the  following  detector: 

if  the  received  element  of  Bn  lies  in  column  i  of  the  array  (4),  the 

detector  prints  the  letter  Ai  ,i  =  1,2,  •  •  •  ,  ju.  The  array  (4)  is  to  (8) 

be  constructed  according  to  the  convention  (5). 

The  following  propositions  and  theorems  can  be  proved  concerning 
signaling  with  an  (n,  /c)-alphabet  and  the  detection  scheme  given  by  (8). 

1.6  BEST  DETECTOR  AND  SYMMETRIC  SIGNALING 

Define  the  probability  /,•  =  ((Ti)  of  an  element  Ti  of  Bn  to  be  A  = 
^wi^n-uf  ^yYiere  p  and  q  are  as  in  (1)  and  Wi  is  the  weight  of  Ti .  Let 


A   CLASS   OF   BINARY   SIGNALING   ALPHABETS  209 

Qi ,  i  =  1 ,  2,  •  •  •  ,  jLi  be  the  sum  of  the  probabilities  of  the  elements  in 
the  iih.  column  of  the  standard  array  (4). 

Proposition  1.  The  probability  that  any  transmitted  letter  of  the 
(n,  A;) -alphabet  be  produced  correctly  by  the  detector  is  Qi  . 

Proposition  2.  The  equivocation^  per  symbol  is 


1    ** 
Hy{x)  =  —  S  Qi  log2  Qi 


n  i=i 

Theorem  1 .  The  detector  (8)  is  a  maximum  likelihood  detector.  That 
is,  for  the  given  alphabet  no  other  detection  scheme  has  a  greater  average 
probability  that  a  transmitted  letter  be  produced  correctly  by  the  de- 
tector. 

Let  us  return  to  the  geometrical  picture  of  w-place  binary  sequences 
as  vertices  of  a  unit  cube  in  n-space.  The  choice  of  a  i^-letter,  n-place 
alphabet  corresponds  to  designating  K  particular  vertices  as  letters. 
Since  the  binary  sequence  corresponding  to  any  vertex  can  be  produced 
by  the  channel  output,  any  detector  must  consist  of  a  set  of  rules  that 
associates  various  vertices  of  the  cube  with  the  vertices  designated  as 
letters  of  the  alphabet.  We  assume  that  every  vertex  is  associated  with 
some  letter.  The  vertices  of  the  cube  are  divided  then  into  disjoint  sets, 
Wi ,  Wi ,  •  •  •  ,  Wk  where  Wi  is  the  set  of  vertices  associated  with  tth 
letter  of  the  signaling  alphabet.  A  maximum  likelihood  detector  is  char- 
acterized by  the  fact  that  every  vertex  in  Wi  is  as  close  to  or  closer  to 
the  iih.  letter  than  to  any  other  letter,  i  =  1,2,  •  •  •  ,  K.  For  group  alpha- 
bets and  the  detector  (8),  this  means  that  no  element  in  the  iih.  column 
of  array  (4)  is  closer  to  any  other  A  than  it  is  to  ^i ,  z  =  1,  2,  •  •  •  ,  ;u. 

Theorem  2.  Associated  with  each  {n,  /(;)-alphabet  considered  as  a  point 
configuration  in  Euclidean  n-space,  there  is  a  group  of  n  X  n  orthogonal 
matrices  which  is  transitive  on  the  letters  of  the  alphabet  and  which 
leaves  the  unit  cube  invariant.  The  maximum  likelihood  sets  1^1  , 
W2 ,  •  •  •  Wn  are  all  geometrically  similar. 

Stated  in  loose  terms,  this  theorem  asserts  that  in  an  (n,  A;)-alphabet 
every  letter  is  treated  the  same.  Every  two  letters  have  the  same  number 
of  nearest  neighbors  associated  with  them,  the  same  number  of  next 
nearest  neighbors,  etc.  The  disposition  of  points  in  any  two  W  regions 
is  the  same. 

1.7  GROUP  ALPHABETS  AND  PARITY  CHECKS 

Theorem  3.  Every  group  alphabet  is  a  systematic^  code:  every  syste- 
matic code  is  a  group  alphabet. 


210  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

We  prefer  to  use  the  word  "alphabet"  in  place  of  "code"  since  the 
latter  has  many  meanings.  In  a  systematic  alphabet,  the  places  in  any 
letter  can  be  divided  into  two  classes :  the  information  places  —  A;  in 
number  for  an  (n,  /c)-alphabet  —  and  the  check  positions.  All  letters 
have  the  same  information  places  and  the  same  check  places.  If  there 
are  k  information  places,  these  may  be  occupied  by  any  of  the  2  /v-place 
binary  sequences.  The  entries  in  the  n  —  k  check  positions  are  fixed 
linear  (mod  2)  combinations  of  the  entries  in  the  information  positions. 
The  rules  by  which  the  entries  in  the  check  places  are  determined  are 
called  parity  checks.  Examples:  for  the  (4,  2)-alphabet  of  (6),  namely 
0000,  1100,  0011,  nil,  positions  2  and  3  can  be  regarded  as  the  informa- 
tion positions.  If  a  letter  of  the  alphabet  is  the  sequence  aia^a^ai ,  then 
ai  =  a2 ,  tti  =  az  are  the  parity  checks  determining  the  check  places  1 
and  4.  For  the  (5,  3)-alphabet  00000,  10001,  01011,  00111,  11010,  10110, 
01100,  11101  places  1,  2,  and  3  (numbered  from  the  left)  can  be  taken 
as  the  information  places.  If  a  general  letter  of  the  alphabet  is  aiazazaiai , 
then  a4  =  a2  -j-  as ,  Ob  =  ai  -j-  a2  -|-  ^3 . 

Two  group  alphabets  are  called  equivalent  if  one  can  be  obtained  from 
the  other  by  a  permutation  of  places.  Example:  the  7  distinct  (3,  2)- 
alphabets  given  in  (3)  separate  into  three  equivalence  classes.  Alpha- 
bets (i),  (ii),  and  (iv)  are  equivalent;  alphabets  (iii),  (v),  (vi),  are  equiva- 
lent; (vii)  is  in  a  class  by  itself. 

Proposition  S.  Equivalent  (n,  fc) -alphabets  have  the  same  probability 
Qi  of  correct  transmission  for  each  letter. 

Proposition  4-  Every  (n,  /c) -alphabet  is  equivalent  to  an  (n,  k)- 
alphabet  whose  first  k  places  are  information  places  and  whose  last  n  —  k 
places  are  determined  by  parity  checks  over  the  first  k  places. 

Henceforth  we  shall  be  concerned  only  with  (n.  A;) -alphabets  w^hose 
first  k  places  are  information  places.  The  parity  check  rules  can  then 
be  written 

k 
ai  =  S  Tij-ay  ,         t  =  /b  -j-  1,  •  •  •  ,  n  (9) 

where  the  sums  are  of  course  mod  2.  Here,  as  before,  a  typical  letter  of 
the  alphabet  is  the  sequence  aia^  •  ■  -  ttn  .  The  jn  are  k(n  —  k)  quantities, 
zero  or  one,  that  serve  to  define  the  particular  (n,  A;)-alphabet  in  question. 

1.8   MAXIMUM  LIKELIHOOD  DETECTION  BY  PARITY  CHECKS 

For  any  element,  J\  of  Bn  we  can  form  the  sum  given  on  the  right  of 
(9).  This  sum  maj^  or  may  not  agree  with  the  symbol  in  the  ?'th  place  of 


A    CLASS   OF   BINARY   SIGNALING   ALPHABETS  211 

T.  If  it  does,  we  say  T  satisfies  the  tth-place  parity  check;  otherwise  T 
fails  the  zth-place  parity  check.  When  a  set  of  parity  check  rules  (9)  is 
giN'cii,  we  can  associate  an  (n  —  /i^-place  binary  sequence,  R{T),  with 
each  element  T  of  5„.  We  examine  each  check  place  of  T  in  order  starting 
with  the  (k  -\-  1  )-st  place  of  T.  We  write  a  zero  if  a  place  of  T  satisfies 
the  parity  check;  we  write  a  one  if  a  place  fails  the  parity  check.  The  re- 
sultant sequence  of  zeros  and  ones,  written  from  left  to  right  is  R(T). 
We  call  R(T)  the  parity  check  sequence  of  T.  Example:  with  the  parity 
rules  04  =  02  -j-  03  ,  05  =  Oi  -j-  02  -j-  c^s  used  to  define  the  (5,  3)-alphabet 
in  the  examples  of  Theorem  3,  we  find  i?(11000)  =  10  since  the  sum  of 
the  entries  in  the  second  and  third  places  of  11001  is  not  the  entry  of 
the  fourth  place  and  since  the  sum  of  Oi  =  1,  02  =  1,  and  03  =  0  is 
0  =  05  . 

Theorem  4-  Let  I,  A2 ,  •  •  •  ^^^  be  an  {n,  /c)-alphabet.  Let  R{T)  be  the 
parity  check  sequence  of  an  element  T  of  B„  formed  in  accordance  with 
the  parity  check  rules  of  the  (n,  /c) -alphabet.  Then  R(Ti)  =  R(T2)  if 
and  only  if  Ti  and  T2  lie  in  the  same  row  of  array  (4).  The  coset  leaders 
can  be  ordered  so  that  R{Si)  is  the  binary  symbol  for  the  integer  i  —  1. 

As  an  example  of  Theorem  4  consider  the  (4,  2)-alphabet  shown  with 
its  cosets  below 


0000 

1011 

0101 

1110 

0100 

nil 

0001 

1010 

0010 

1001 

0111 

1100 

1000 

0011 

1101 

0110 

The  parity  check  rules  for  this  alphabet  are  03  =  oi  ,  04  =  Oi  -j-  ^2  • 
Every  element  of  the  second  row  of  this  array  satisfies  the  parity  check 
in  the  third  place  and  fails  the  parity  check  in  the  4th  place.  The  parity 
check  sequence  for  the  second  row  is  01.  The  parity  check  for  the  third 
row  is  10,  and  for  the  fourth  row  11.  Since  every  letter  of  the  alphabet 
satisfies  the  parity  checks,  the  parity  check  sequence  for  the  first  row  is 
00.  We  therefore  make  the  following  association  between  parity  check 
sequences  and  coset  leaders 

00  -^  0000   =   Si 

01  -^  0100   =   S2 

10  -^  0010  =   S, 

11  -^  1000  =   ^4 

1.9  INSTRUMENTING   A    GROUP   ALPHABET 

Proposition  4  attests  to  the  ease  of  the  encoding  operation  involved 


212  THE   BELL   SYSTEM  TECHNICAL   JOURNAL,    JANUARY    1956 

with  the  use  of  an  (n,  fc) -alphabet.  If  the  original  message  is  presented  as 
a  long  sequence  of  zeros  and  ones,  the  sequence  is  broken  into  blocks  of 
length  k  places.  Each  block  is  used  as  the  first  k  places  of  a  letter  of 
the  signaling  alphabet.  The  last  n-k  places  of  the  letter  are  determined 
by  fixed  parity  checks  over  the  first  k  places. 

Theorem  4  demonstrates  the  relative  ease  of  instrumenting  the  maxi- 
mum hkelihood  detector  (8)  for  use  with  an  (n.  A:) -alphabet.  When  an 
element  T  of  Bn  is  received  at  the  channel  output,  it  is  subjected  to  the 
n-k  parity  checks  of  the  alphabet  being  used.  This  results  in  a  parity 
check  sequence  R{T).  R(T)  serves  to  identify  a  unique  coset  leader,  say 
Si .  The  product  SiT  is  then  formed  and  produced  as  the  detector  out- 
put. The  probability  that  this  be  the  correct  letter  of  the  alphabet  is  Qi  . 

1.10   BEST   GROUP   ALPHABETS 

Two  important  questions  regarding  (n,  fc)-alphabets  naturally  arise. 
What  is  the  maximum  value  of  Qi  possible  for  a  given  n  and  k  and  which 
of  the  N(n,  k)  different  subgroups  give  rise  to  this  maximum  Qi?  The 
answers  to  these  questions  for  general  n  and  k  are  not  known.  For  many 
special  values  of  n  and  k  the  answers  are  known.  They  are  presented  in 
Tables  II,  III  and  IV,  which  are  explained  below. 

The  probability  Qi  that  a  transmitted  letter  be  produced  correctly  by 
the  detector  is  the  sum,  Qi  =  ^i  f{Si)  of  the  probabilities  of  the  coset 
leaders.  This  sum  can  be  rewritten  as  Qi  =  2Zi=o  ««  P^Q^~^  where  a,  is 
the  number  of  coset  leaders  of  weight  i.  One  has,  of  course,  ^a,  =  v  = 

/  y)  \  T?  ' 

2^"''  for  an  (n,  /(;)-alphabet.  Also  «>  ^  (  .  )  =  -7-7 — '■ — n- !  since  this  is  the 

\t  /       tlin  —  t) 

number  of  elements  of  Bn  of  weight  i. 

The  (Xi  have  a  special  physical  significance.  Due  to  the  noise  on  the 
channel,  a  transmitted  letter,  A,  ,  of  an  (n,  /c)-alphabet  will  in  general  be 
received  at  the  channel  output  as  some  element  T  of  Bn  different  from 
Ai  .li  T  differs  from  Ai  in  s  places,  i.e.,  if  w{AiT)  =  s,  we  say  that  an 
s-tuple  error  has  occurred.  For  a  given  (n,  fc)-alphabet,  ai  is  the  number 
of  i-tuple  errors  which  can  be  corrected  by  the  alphabet  in  question, 
i  =  0,  1,2,  ■  •  •  ,  n. 

Table  II  gives  the  a{  corresponding  to  the  largest  possible  value  of  Qi 
for  a  given  k  and  ?i  for  k  =  2,3,  •••w—  l,n  =  4---  ,10  along  with  a 
few  other  scattered  values  of  n  and  k.  For  reference  the  binomial  coeffi- 
cients (  .  )  are  also  listed.  For  example,  we  find  from  Table  II  that  the 
best  group  alphabet  with  2    =16  letters  that  uses  n  =  10  places  has  a 


A    CLASS    OF   BINARY   SIGNALING   ALPHABETS  213 

1 A  Q  C        'J  **        Q 

probability  of  correct  transmission  Qi  =  q  +  lOg  p  +  39g  p"  +  l-Ag'p  . 
The  alphabet  corrects  all  10  possible  single  errors.  It  corrects  39  of  the 

possible  f  .^  j  =  45  double  errors  (second  column  of  Table  II)  and  in 

addition  corrects  14  of  the  120  possible  triple  errors.  By  adding  an  addi- 
tional place  to  the  alphabet  one  obtains  with  the  best  (11,  4)-alphabet 
an  alphabet  with  16  letters  that  corrects  all  11  possible  single  errors  and 
all  55  possible  double  errors  as  well  as  61  triple  errors.  Such  an  alphabet 
might  be  useful  in  a  computer  representing  decimal  numbers  in  binary 
form. 

For  each  set  of  a's  listed  in  Table  II,  there  is  in  Table  III  a  set  of 
parity  check  rules  which  determines  an  {n,  A)-alphabet  having  the  given 
a's.  The  notation  used  in  Table  III  is  best  explained  by  an  example.  A 
(10,  4)-alphabet  which  realizes  the  a's  discussed  in  the  preceding  para- 
graph can  be  obtained  as  follows.  Places  1,  2,  3,  4  carrj-  the  information. 
Place  5  is  determined  to  make  the  mod  2  sum  of  the  entries  in  places 
3,  4,  and  5  ecjual  to  zero.  Place  6  is  determined  by  a  similar  parity  check 
on  places  1,  2,  3,  and  6;  place  7  by  a  check  on  places  1,  2,  4,  and  7,  etc. 

It  is  a  surprising  fact  that  for  all  cases  investigated  thus  far  an  {n,  k)- 
alphabet  best  for  a  given  value  of  p  is  uniformly  best  for  all  values  of 
p,  0  ^  p  ^  1 2.  It  is  of  course  conjectured  that  this  is  true  for  all  n  and  /,-. 

It  is  a  further  (perhaps)  surprising  fact  that  the  best  {n,  fc) -alphabets 
are  not  necessarily  those  with  greatest  nearest  neighbor  distance  be- 
tween letters  when  the  alphabets  are  regarded  as  point  configurations  on 
the  n-cube.  For  example,  in  the  best  (7,  3)-alphabet  as  listed  in  Table 
III,  each  letter  has  two  nearest  neighbors  distant  3  edges  away.  On  the 
other  hand,  in  the  (7,  3)-alphabet  given  by  the  parity  check  rules  413, 
512,  623,  7123  each  letter  has  its  nearest  neighbors  4  edges  away.  This 
latter  alphabet  does  not  have  as  large  a  value  of  Qi ,  however,  as  does 
the  (7,  3)-alphabet  listed  on  Table  III. 

The  cases  /.;  =  0,  1,  /?  —  1,  n  have  not  been  listed  in  Tables  II  and  III. 
The  cases  k  =  0  and  k  =  n  are  completely  trivial.  For  k  =  1,  all  n  >  1 
the  best  alphabet  is  obtained  using  the  parity  rule  a>  =  03=  •  •  •  = 
a„  =  oi  .  If  n  =  '2j, 

If  n  =  2j  +  1,  Qi  =  i:  (^')  pY-\ 

For  k  =  n  —  1,  /;  >  1.  the  maximum  Qi  is  Qi  =  g"~  and  a  parity  rule 
for  an  alphabet  realizing  this  Qi  is  o„  =  oi . 

If  the  a's  of  an  (/<,  A)-alphabet  are  of  the  form  a,  =  (  .  j ,  i  =  0,  1, 


214  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 


Table  II  —  Probability  of  No  Error  with  Best 
Alphabets,  Qi  =   2Z  «»P*2"~' 


(?) 

k   =  2 

k  =   3 

k  =   4 

k  =  5 

k  =  6 

k   =  7 

k  =  8 

k  =  9 

*  =  10 

i 
0 

ai 

(li 

a 

a, 

ai 

Oi 

fli 

ai 

a; 

n  =  4 

1 

1 

1 

4 

3 

0 

1 

1 

1 

n  =   5 

1 

2 
0 

5 
10 

1 

5 
2 

1 

3 

1 

1 

71  =  6 

1 
2 

6 
15 

6 
9 

6 
1 

3 

0 

1 

1 

1 

1 

1 

n  =  7 

1 
2 
3 

7 
21 
25 

7 

18 

6 

7 
8 

7 

3 

0 

1 

1 

1 

1 

1 

1 

n   =  8 

1 
2 
3 

8 
28 
56 

8 
28 
27 

8 

20 

3 

8 

7 

7 

3 

0 

1 

1 

1 

1 

1 

1 

1 

1 

9 

9 

9 

9 

9 

7 

3 

n  =  9 

2 
3 

4 

36 

84 

126 

36 
64 

18 

33 
21 

22 

6 

0 

1 

1 

1 

1 

1 

1 

1 

1 

1 

10 

10 

10 

10 

10 

10 

7 

3 

n  =  10 

2 
3 

4 

45 
120 
210 

45 

110 

90 

45 
64 

8 

39 
14 

21 

5 

0 

1 

1 

1 

1 

1 

1 

1 

1 

1 

11 

11 

11 

11 

11 

11 

7 

3 

n  =  11 

2 
3 
4 
5 

55 
165 
330 

462 

55 
165 
226 

54 

55 

126 

63 

55 

61 

20 

4 

0 

1 

1 

1 

1 

1 

1 

1 

1 

12 

12 

12 

12 

12 

7 

3 

n  =  12 

2 
3 
4 
5 

66 
220 
495 
792 

66 
220 
425 
300 

66 
200 
233 

19 

3 

A    CLASS    OF   BINARY   SIGNALIHG    ALPHABETS  215 

2,  •  •  •  ,  j,  «j+i  =  f  some  integer,  aj+o  =  ay+s  =  •  •  •  =  «„  =  0,  then 
there  does  not  exist  a  2  -letter,  w-place  alphabet  of  any  sort  better  than 
the  given  (n,  A)-alphabet.  It  will  be  observed  that  many  of  the  a's  of 
Table  II  are  of  this  form.  It  can  be  shown  that 

Proposition  5  ii  n  -\-  I       „       /"t"!       q       1^2"^*  —  1  there  exists 

no  2'''-letter,  n-place  alphabet  better  than  the  best  (n,  /c) -alphabet. 
When  the  inequality  of  proposition  5  holds  the  a's  are  either  «o  =  1, 

""''  -  1,  all  other  «  =  0;  or  ao  =  1,  «i  =  (Vj  ,  «2  =  2"~'  -  1  - 

,  all  other  a  =  0;  or  the  trivial  ao  =  1  all  other  a  =  0  which  holds 

uhen  k  =  n.  The  region  of  the  n  —  k  plane  for  which  it  is  known  that 
(n,  A-)-alphabets  cannot  be  excelled  by  any  other  is  shown  in  Table  IV. 

1.11    A   DETAILED    EXAMPLE 

As  an  example  of  the  use  of  {n,  A") -alphabets  consider  the  not  un- 
realistic case  of  a  channel  with  -p  =  0.001,  i.e.,  on  the  average  one  binary 
digit  per  thousand  is  received  incorrectly.  Suppose  we  wish  to  transmit 
messages  using  32  different  letters.  If  we  encode  the  letters  into  the  32 
5-place  binary  sequences  and  transmit  these  sequences  without  further 
encoding,  the  probability  that  a  received  letter  be  in  error  is  1  — 
(1  _  pf  =  0.00449.  If  the  best  (10,  5)-alphabet  as  shown  in  Tables  II 
and  III  is  used,  the  probability  that  a  letter  be  wrong  is  1  —  Qi  = 
1  -  r/"  -  lOgV  -  21gy  -  24/)'  -  72p'  +  •  •  •  =  0.000024.  Thus 
by  reducing  the  signaling  rate  by  ^^,  a  more  than  one  hundredfold  re- 
duction in  probability  of  error  is  accomplished. 

A  (10,  5)-alphabet  to  achieve  these  results  is  given  in  Table  III.  Let 
a  typical  letter  of  the  alphabet  be  the  10-place  sequence  of  binary  digits 
aia2  ■  •  •  agttio  .  The  symbols  aia^Ozaia^  carry  the  information  and  can  be 
any  of  32  different  arrangements  of  zeros  and  ones.  The  remaining  places 
are  determined  by 

06  =  ai  -j-  a-i  -j-  a4  -j-  ^5 

a?  =  tti  -j-  oo  -f  a4  -j-  as 

as  =  ai  -j-  a2  +  a.3  +  Os 

ag  =  Oi  +  02  4-  Qi  -j-  0,4 

Oio  =  Oi  +  a-i  -j-  03  4-  04  4-  «5 

To  design  the  detector  for  this  alphabet,  it  is  first  necessary  to  deter- 
mine the  coset  leaders  for  a  standard  array  (4)  formed  for  this  alphabet. 


•Jl 

t-l 

a 
pa 
< 
M 

Ph 
< 

cc 

o 

H 


O 

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ti; 
O 

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I— I 

-< 
Ph 


P3 


t^  00 


-f  ^  cc 

CC  C^)  !M 


O  t^  X 


lO  a;  t^  oc 


00  C2 


^  ^  CC 
CC  (N  C^l 


t-  GC 


lO  ic  lO  -r 

-f  -^  CC  CT 
CC  C^  CM  C^I 

;C  1^  X  c: 


^ 

cc  -+ 

-f  -^  cc 

-f  -^  cc  cc 

-r  -^  cc  re  cc 

(M  <N 

CC  C^  CM 

CC  CM  CM  CM 

re  T-l  CM  CM  CM 

ic  :c  I-  y:  — 


i 


re  cc 

CO 

ce 

C^l  cc 

CM  cc  re 

re  C^J  CM  CM 

CM  re  re  c^i 

CM  re  re  CM 

^—    .-H 

.-H  -—  C^l 

_  ,_  —  ,-H 

r—  ^-  T-H  (M  ,— . 

T-^  CM  ^  ^  CM  -^ 

'^^  lo 

•^  lO  « 

-*  iC  <£)  t^ 

•^  lO  CO  t^  oc 

"*  >OCD  t^OO  C5 

C^l 


ex 

re 


C^l  CM  C^l 

re-rocot^      ce-^iocot^oc 


C^l  C^l 


'  >o 


re  f  lO  CO 


re  T  lO  CO  t^  oC' 


iCi 


CO 


oc 


210 


1—1  1—1 

^CM 

1-H  1-H 

cO'f  -* 

CM  CM  CO 

1-1 1-l 

T— 1  1-H  1-H 

Ot-h 

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1—1 1—1 

I-H  1-H  1-H 

00 

^cot-oo 

^^iCiO 

134 

0  124 

1  123 

12351 

0  123 

1  124 
2134 

Ol  ^ 

01  .—1 1—1 

"^  1-H  1-H  1-H 

r^ 

t-  t-  t^ 

coco 

CO  CO  lO 

•^iO-* 

^^10^-^ 

CO 

"^^00  CO 

"''^  CO  CO  CO 

-*  't<  jvj 

^'*(N^ 

'^'*  CM  CM  CM 

CO(M^ 

«^^^ 

COCM^^^ 

^-^o 

-^-^O-H 

^'~'  0--HCM 

GOO-J  T-1 

00  01 1-1 1—1 

00  02  1-H  1-H  1-H 

CO 

CO  iC 

^-t 

OCOCO^^ 

iOiO>Oj^ 

'^'^'^coco 

-^-^^0^ 

'^'^^cacM 

CO!M  C^  ^ 

CO  <M  CM  ^  ^ 

T— I   1-H   T-H    _^ 

1— 1  1 — 1  r— *    -^ 

o 

C  "—1 

t^  00  o  i-H 

t^  00  C5  1— 1  >— 1 

iCi 

'I* 

lOiOiO'*^  j^ 

•r-^  c^cc  ^ 

CO  CM  iM  O)  ^ 

CO  t^  00  02  ^ 

•* 

^ 

CO 

^co 

CO^  ^  -* 

CM 

'^'cOCM 

'f  CM  (M  CO  CO 

CO  ^  "*  coco  j^ 

1— H 

CO   T-H   1— 1   ^H   C^l 

0 

1— 1  CM  r-(  CM  .-<  ^ 

1— 1 

10  CO  t^  00  cr. 

10  CO  t^  00  0  rH 

1-H 

CO 

coco 

CO  CO 

CM 

(N  (M 

CO  CM  CM 

CI  CO  CO 

r— 1 

coco  CM 

T—l 

T~i 

o*o*c<i^^^ 

1— 1   CM   CO   r- 1   .— 1 

CM 

0 

1— ( 

CO  CO  CM  1-1  1—1  1—1 

0 

1-H 

^  CM  CO  -1  1-H  -H  ^  ^  ^ 

^lOfOi^ccoi 

•^  lO  CO  t^  00  Ci 

I— 1 

1— 1 

rflOCOt^OOOli-Hi-Hi-H 

CM 

C^  C^l 

CM  CM  CM 

CM  (M 

!M 

T-H 

1—1 

1— t 

t— 1  1 — I 

1— 1  ^H  I— 1  CM  CM 

1-H 

T— t 

0 

r-i  .-<  i-H  CM  CM  CM 

1— t 

0 

1-H 

^  ^  ^  ^  CM  CM  CM  ^  ^  ^ 

CO  'tl  lO  CO  t^  00  05 

>— 1 

CO  ■*  10  CO  t>  00  Oi 

i-H 

r— ( 

CO-*lOCOt^00C2l-Hr-l^ 

0 

I-l 

CM 

T— 1 

1— ( 

1— t 

II 

II 

II 

e 

e 

e 

217 


218  THE    BELL    SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

Table  IV  —  Region  of  the  n-k  Plane  for  Which  it  is  Known 

THAT    [n,    fc)-ALPHABETS    CaNNOT   Be   EXCELLED 

k 

30 

29 

28  •  •  • 

27  .... 

26  

25  

24  

23  

22  

21  

20  

19  

18  

17  

16  

15  

14  

13  

12  

11  

10  

9  

8  

7  

6         

5        

4      

3     

2   

1 


\ 


0  1  2  3  4  5  6  7  8  9  10  12  14  16  18  20  22  24  26  28  30  n 

This  can  be  done  by  a  \'ariety  of  special  methods  which  considerably 
reduce  the  obvious  labor  of  making  such  an  array.  A  set  of  best  »S's  along 
with  their  parity  check  symbols  is  given  in  Table  V. 

A  maximum  likelihood  detector  for  the  (10,  5)-alphabet  in  question 
forms  from  each  received  sequence  6162  •  •  •  &10  the  parity  check  symbol 
C1C2C3C4C5  where 

Ci  =  h  4-  ^h   4-  ^3  +  Ih  +  ^5 

C2  =  67  -(-  6i  -]-  h-i  +  hi  \-  Ih 

Cs  =  &8  +  ^^1  +  h  -j-  Ih  +  ^5 

Ci  =  bg  +  hi   4-  h-i  -i-  h-.i  -(-  hi 

C5   =    />in  +  hi  +   />,  +  h,  4-  hi  4-  65 

According  to  Table  V,  if  CiC-jCiAf'b  contains  less  than  three  ones,  the  de- 
tector should  brint  hih^kihih^  .  The  detector  should  piint  (/m  4-  1)^2^3^4'':. 
if  the  parity  check  sequence  C1C2C3C4C5  is  either  11111  oi-  11110;  the  dv- 


A    CLASS    OF    BINARY    SIGNALING    ALPHABETS 


219 


Table  V  —  Coset  Leaders  and  Parity  Check  Sequences 

FOR  (10, 5) -Alphabet 


ClCiCsCiCb 

^        s 

CIC2C3C4C6 

5 

00000 

0000000000 

11100 

0000100001 

10000 

0000010000 

11010 

0001000001 

01000 

0000001000 

11001 

0001000010 

00100 

0000000100 

10110 

0010000001 

00010 

0000000010 

10101 

0010000010 

00001 

0000000001 

10011 

OOIOOOOIOO 

1 1000 

0000011000 

OHIO 

0100000001 

10100 

0000010100 

01101 

0100000010 

10010 

0000010010 

01011 

0100000100 

10001 

0000010001 

00111 

0100001000 

01100 

0000001100 

11110 

1000000001 

01010 

0000001010 

11101 

OOOOIOOOOO 

01001 

0000001001 

11011 

OOOIOOOOOO 

00110 

0000000110 

10111 

0010000000 

00101 

0000000101 

01111 

0100000000 

00011 

0000000011 

mil 

1000000000 

tector  should  print  61(62  -j-  l)b3lhh^  if  the  parity  check  sequence  is  01111, 
00111,  01011,  01101,  or  OHIO;  the  detector  should  print  hMb-i  +  1)6465 
if  the  parity  check  sequence  is  10111,  10011,  10101,  or  10110;  the  de- 
tector should  print  616263(64  -j-  1)65  if  the  parity  check  sequence  is  11011, 
11001,  11010;  and  finally  the  detector  should  print  61626364(65  -j-  1)  if  the 
parity  check  sequence  is  11101  or  11100. 

Simpler  rules  of  operation  for  the  detector  may  possibly  be  obtained 
by  choice  of  a  different  set  of  S's  in  Table  V.  These  quantities  in  general 
are  not  unique.  Also  there  may  exist  non-equivalent  alphabets  with 
simpler  detector  rules  that  achieve  the  same  probability  of  error  as  the 
alphabet  in  question. 


I'vrt  II  —  Additional  Theory  and  Proofs  of  Theorems  of  Part  I 

'  2.1  the  abstract  group  Cn 

It  will  be  helpful  here  to  say  a  few  more  words  about  Br,  ,  the  group 

of  n-place  binary  sequences  under  the  operation  of  addition  mod  2.  This 

j  group  is  simply  isomorphic  with  the  abstract  group  Cn  generated  by  n 

\  commuting  elements  of  order  two,  say  ai,    a-2  ,  ■  ■  ■  ,  a„  .  Here  a,:ay  = 

<i,ai  and  a/  =  /,  i,  j  =  1,  2,  •  •  •  ,  n,  where  /  is  the  identity  for  the 

group.  The  eight  distinct  elements  of  C3  are,  for  example,  /,  o-i  ,  a-y , 

(h  ,  (iici-,  ,  aio-.i  ,  a-itti  ,  aia-ittz .  The  group  C„  is  easily  seen  to  be  isomorphic 

I  with  the  Ai-fold  direct  product  of  the  group  Ci  with  itself. 


220  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

It  is  a  considerable  saving  in  notation  in  dealing  with  C„  to  omit  the 
symbol  "a"  and  write  only  the  subscripts.  In  this  notation  for  example, 
the  elements  of  d  are  7,  1,  2,  3,  4,  12,  13,  14,  23,  24,  34,  123,  124,  134, 
234,  1234.  The  product  of  two  or  more  elements  of  C„  can  readily  be 
written  down.  Its  symbol  consists  of  those  numerals  that  occur  an  odd 
number  of  times  in  the  collection  of  numerals  that  comprise  the  sym- 
bols of  the  factors.  Thus,  (12)(234)(123)  =  24. 

The  isomorphism  between  Cn  and  Bn  can  be  established  in  many  ways. 
The  most  convenient  way,  perhaps,  is  to  associate  with  the  element 
iii-2H  ■  ■  ■  ik  of  Cn  the  element  of  Bn  that  has  ones  in  places  ii  ,1-2,  •  •  •  ,  ik 
and  zeros  in  the  remaining  n  —  k  places.  For  example,  one  can  associate 
124  of  C4  with  1101  of  Bi  ;  14  with  1001,  etc.  In  fact,  the  numeral  no- 
tation afforded  by  this  isomorphism  is  a  much  neater  notation  for  Bn 
than  is  afforded  by  the  awkward  strings  of  zeros  and  ones.  There  are, 
of  course,  other  ways  in  which  elements  of  C„  can  be  paired  with  elements 
of  Bn  so  that  group  multiplication  is  preserved.  The  collection  of  all  such 
"pairings"  makes  up  the  group  of  automorphisms  of  C„ .  This  group  of 
automorphisms  of  Cn  is  isomorphic  with  the  group  of  non-singular  linear 
homogenous  transformations  in  a  field  of  characteristic  2. 

An  element  T  of  C„  is  said  to  be  dependent  upon  the  set  of  elements 
Ti ,  T2 ,  •  •  ■  ,  Tj  oi  Cn  if  T  can  be  expressed  as  a  product  of  some  ele- 
ments of  the  set  Ti  ,  T2 ,  •  •  •  ,  Tj ;  otherwise,  T  is  said  to  be  independent 
of  the  set.  A  set  of  elements  is  said  to  be  independent  if  no  member  can 
be  expressed  solely  in  terms  of  the  other  members  of  the  set.  For  example, 
in  Cs ,  1,  2,  3,  4  form  a  set  of  independent  elements  as  do  likewise  2357, 
12357,  14.  However,  135  depends  upon  145,  3457,  57  since  135  = 
(145)  (3457) (57).  Clearly  any  set  of  n  independent  elements  of  Cn  can 
be  taken  as  generators  for  the  group.  For  example,  all  possible  products 
formed  of  12,  123,  and  23  yield  the  elements  of  C3  . 

Any  k  independent  elements  of  C„  serve  as  generators  for  a  subgroup 
of  order  2*".  The  subgroup  so  generated  is  clearly  isomorphic  with  Ck  ■ 
All  subgroups  of  C„  of  order  2''  can  be  obtained  in  this  way. 

The  number  of  ways  in  which  k  independent  elements  can  be  chosen 
from  the  2"  elements  of  C„  is 

F{n,  k)  -  (2"  -  2'')(2"  -  2')(2"  -  2')  •  •  •  (2"  -  2'-') 

For,  the  first  element  can  be  chosen  in  2"  —  1  ways  (the  identity  cannot 
be  included  in  a  non-trivial  set  of  independent  elements)  and  the  second 
element  can  be  chosen  in  2"  —  2  ways.  These  two  elements  determine  a 
subgroup  of  order  2\  The  third  element  can  be  chosen  as  any  element  of 
the  remaining  2"  —  2"  elements.  The  3  elements  chosen  determine  a 


I 


A    CLASS    OF    BINARY   SIGNALING   ALPHABETS  221 

subgroup  of  order  2l  A  fourth  independent  element  can  be  chosen  as 
any  of  the  remaining  2"  —  2  elements,  etc. 

Each  set  of  k  independent  elements  serves  to  generate  a  subgroup  of 
order  2''.  The  quantity  F{n,  k)  is  not,  however,  the  number  of  distinct 
subgroups  of  C„  of  this  order,  for,  a  given  subgroup  can  be  obtained 
from  many  different  sets  of  generators.  Indeed,  the  number  of  different 
sets  of  generators  that  can  generate  a  given  subgroup  of  order  2^  of  C„ 
is  just  F{k,  k)  since  any  such  subgroup  is  isomorphic  with  Ck  .  Therefore 
the  number  of  subgroups  of  Cn  of  order  2''  is  N{n,  k)  =  F(n,  k)/F(k,  k) 
which  is  (2).  A  simple  calculation  gives  N(n,  k)  =  N(n,  n  —  k). 

2.2    PROOF   OF   PROPOSITIONS    1    AND   2 

After  an  element  A  of  5„  has  been  presented  for  transmission  over 
a  noisy  binary  channel,  an  element  T  of  5„  is  produced  at  the  channel 
output.  The  element  U  =  AT  oi  Bn  serves  as  a  record  of  the  noise 
during  the  transmission.  U  is  an  n-place  binary  sequence  with  a  one  at 
each  place  altered  in  A  by  the  noise.  The  channel  output,  T,  is  obtained 
from  the  input  A  by  multiplication  by  U:  T  =  UA.  For  channels  of  the 
sort  under  consideration  here,  the  probability  that  U  be  any  particular 
element  of  Bn  of  w^eight  w  is  p^'g"""'. 

Consider  now  signaling  with  a  particular  (n,  /b) -alphabet  and  consider 
the  standard  array  (4)  of  the  alphabet.  If  the  detection  scheme  (8)  is 
used,  a  transmitted  letter  A  i  will  be  produced  without  error  if  and  only 
if  the  received  symbol  is  of  the  form  SjAi .  That  is,  there  will  be  no 
error  only  if  the  noise  in  the  channel  during  the  transmission  of  Ai  is 
represented  by  one  of  the  coset  leaders.  (This  applies  (or  i  =  1,2,  •  •  •  , 
fi  =  2  ).  The  probability  of  this  event  is  Qi  (Proposition  1,  Section  1.6). 
The  convention  (5)  makes  Qi  as  large  as  is  possible  for  the  given  alpha- 
bet. 

Let  X  refer  to  transmitted  letters  and  let  Y  refer  to  letters  produced 
by  the  detector.  We  use  a  vertical  bar  to  denote  conditions  when  writing 
probabilities.  The  quantity  to  the  right  of  the  bar  is  the  condition.  We 
suppose  the  letters  of  the  alphabet  to  be  chosen  independently  with 
ec^ual  probability  2"  . 

The  equivocation  h{X  \  Y)  obtained  when  using  an  (n,  fc)-alphabet 
with  the  detector  (8)  can  most  easily  be  computed  from  the  formula 

h(X  I  F)  =  h{X)  -  h(Y)  +  h(Y  I  X)  (10) 

The  entropy  of  the  source  is /i(X)  =  k/n  bits  per  symbol.  The  probability 
that  the  detector  produce  Aj  when  Ai  was  sent  is  the  probability  that 
the  noise  be  represented  by  AiAjSt ,  ^  =  1,2,  •  •  •  ,  v.  In  symbols, 


222  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

Pr{Y  -.  Ai  I  X  -^  Ad  =  Z  Pr{N  -^  AiA.Sc)  =  QiA^A,) 

where  Q{Ai)  is  the  sum  of  the  prol)abiUties  of  the  elements  that  are  in 
the  same  column  as  Ai  in  the  standard  array.  Therefore 

Pr{Y  ->  .4,)  =  E  Pr{Y  ->  A,  \  X  -^  AdPr{X  -^  A^  =  ^  E  QU,A,) 

=  4,        since  E  Q^A.A^  =  E  QUi)  =  1. 
This  last  follows  from  the  group  property  of  the  alphabet.  Therefore 

/i(lO  =  --  E  P>iy  -^  A,)  log  Pr{Y  -^  A,)  =  -  bits/symbol. 
n  n 

It  follows  then  from  (10)  that 

h{X  I  Y)  =  h(Y  I  X) 

The  computation  of  h(Y  \  X)  follows  readily  from  its  definition 

h{Y  I  X)  =  E  Prix  -^  AdhiY  \  X  -^  Ai) 

i 

=  -E  Prix  ->  AdPriY  -^  Aj  \  X  ->  Ai) 

log  PHY -^Aj  I  X-^Ai) 
=  -^,1211  PriN  ->AiScAj)  log  E  PriN  ->  AiS„,Aj) 


I 


=  -^,ZQiAiAj)'}ogQiAiAj) 

Zi       ij 

=  -  EQU,)logQ(A,) 

i 

Each  letter  is  n  binary  places.  Proposition  2,  then  follows. 

2.3   DISTANCE   AND   THE   PROOF   OF   THEOREM    1 

Let  A  and  B  be  two  elements  of  Bn  ■  We  define  the  distance,  diA,  B), 
between  A  and  B  to  be  the  weight  of  their  product, 

d{A,  B)  =  w(AB)  (11) 

The  distance  between  .4  and  B  is  the  number  of  places  in  which  A  and 
B  difTer  and  is  jnsl  the  "Hamming  distance."  ^  In  terms  of  the  n-cube, 
diA,  B)  is  Ihe  minimum  mmiber  of  edges  that  must  be  traversed  to  go 


A    CLASS   OF   BINARY   SIGNALING   ALPHABETS  223 

from  vertex  ^4  to  vertex  B.  The  distance  so  defined  is  a  monotone  fnne- 
tion  of  the  Euchdean  distance  between  vertices. 

It  follows  from  (11)  that  if  C  is  any  element  of  B„  then 

d{A,B)  =  cJ(A(\BC)  (12) 

This  fact  shows  the  detection  scheme  (8)  to  be  a  maximum  likelihood 
detector.  By  definition  of  a  standard  array,  one  has 

d(Si ,  I)  ^  d(S,Aj ,  I)  for  all  i  and  j 

The  coset  leaders  were  chosen  to  make  this  true.  From  (12), 

d(S,  ,  I)  =  d(SiA,„S,- ,  /  .4„.^S,)  =  d(SiA,n ,  A,„) 

d(SAj ,  /)  -  diS^AjSiAm  ,  I  SiAJ  =  diAjA,n ,  SiAr.) 

=  d{SiAm  ,  A() 

where  Af  =  AjA^  .  Substituting  these  expressions  in  the  inecjuality 
above  yields 

d(SiAm  ,  A„,)  ^  d(SiAm  ,  At)  for  all  i,  m,  I 

This  equation  says  that  an  arbitrary  element  in  the  array  (4)  is  at  least 
as  close  to  the  element  at  the  top  of  its  column  as  it  is  to  any  other  letter 
of  the  alphabet.  This  is  the  maximum  likelihood  property. 

2.4   PROOF   OF   THEOREM   2 

Again  consider  an  (n,  /c) -alphabet  as  a  set  of  vertices  of  the  unit  n-cube. 
Consider  also  n  mutually  perpendicular  hyperplanes  through  the  cen- 
troid  of  the  cube  parallel  to  the  coordinate  planes.  We  call  these  planes 
"symmetr}^  planes  of  the  cube"  and  suppose  the  planes  numbered  in 
accordance  with  the  corresponding  parallel  coordinate  planes. 

The  reflection  of  the  vertex  with  coordinates  (ai  ,  a^ ,  •  •  •  ,  a^ ,  •  •  •  ,  a,j) 
in  symmetry  plane  i  yields  the  vertex  of  the  cube  whose  coordinates 
are  (ai  ,  oo ,  ■  •  •  ,  a,  -j-  1,  •  •  •  ,  0,0 .  More  generally,  reflecting  a  given 
vertex  successively  in  symmetry  planes  i,  j,  k,  ■  •  ■  yields  a  new  vertex 
whose  coordinates  differ  from  the  original  vertex  precisely  in  places 
i,  j,  k  ■  ■  ■  .  Successive  reflections  in  hyperplanes  constitute  a  transfor- 
mation that  leaves  distances  between  points  unaltered  and  is  therefore 
a  "rotation."  The  rotation  obtained  by  reflecting  successively  in  sym- 
metry planes  ?',  j,  k,  etc.  can  be  represented  by  an  ?i-place  symbol  having 
a  one  in  places  ?',  j,  k,  etc.  and  a  zero  elsewhere. 

We  now  regard  a  given  {n,  /j)-alphabet  as  generated  by  operating  on 
the  vertex  (0,  0,  •  •  ■  ,  0)  of  the  cube  with  a  certain  collection  of  2    ro- 


224  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


tation  operators.  The  symbols  for  these  operators  are  identical  with  the 
sequences  of  zeros  and  ones  that  form  the  coordinates  of  the  2  points. 
It  is  readily  seen  that  these  rotation  operators  form  a  group  which  is 
transitive  on  the  letters  of  the  alphabet  and  which  leave  the  unit  cube 
invariant.  Theorem  2  then  follows. 

Theorem  2  also  follows  readily  from  consideration  of  the  array  (4). 
For  example,  the  maximum  likelihood  region  associated  with  /  is  the 
set  of  points  I,  So ,  S3 ,  •  •  •  ,  Sy  .  The  maximum  likelihood  region  asso- 
ciated with  A;  is  the  set  of  points  Ai ,  AiS^ ,  AiSs ,  ■  •  ■  ,  AiSy .  The 
rotation  (successive  reflections  in  symmetry  planes  of  the  cube)  whose 
symbol  is  the  same  as  the  coordinate  sequence  of  Ai  sends  the  maximum 
likelihood  region  of  /  into  the  maximum  likelihood  region  oi  Ai ,  i  = 
1,  2,  • •  •  ,  M. 

2.5  PROOF    OF   THEOREM   3 

That  every  systematic  alphabet  is  a  group  alphabet  follows  trivially 
from  the  fact  that  the  sum  mod  2  of  two  letters  satisfying  parity  checks 
is  again  a  letter  satisfying  the  parity  checks.  The  totality  of  letters  satis- 
fying given  parity  checks  thus  constitutes  a  finite  group. 

To  prove  that  every  group  alphabet  is  a  systematic  code,  consider 
the  letters  of  a  given  (w,  /c) -alphabet  listed  in  a  column.  One  obtains  in 
this  way  a  matrix  with  2  rows  and  n  columns  whose  entries  are  zeros 
and  ones.  Because  the  rows  are  distinct  and  form  a  group  isomorphic  to 
Ck  ,  there  are  k  linearly  independent  rows  (mod  2)  and  no  set  of  more 
than  h  independent  rows.  The  rank  of  the  matrix  is  therefore  h.  The 
matrix  therefore  possesses  k  linearly  independent  (mod  2)  columns  and 
the  remaining  n  —  k  columns  are  linear  combinations  of  these  A;.  Main- 
taining only  these  k  linearly  independent  columns,  we  obtain  a  matrix  of 
k  columns  and  2*'  rows  with  rank  k.  This  matrix  must,  therefore,  have  k 
linearly  independent  rows.  The  rows,  however,  form  a  group  under  mod 
2  addition  and  hence,  since  k  are  linearly  independent,  all  2"  rows  must 
be  distinct.  The  matrix  contains  only  zeros  and  ones  as  entries;  it  has  2 
distinct  rows  of  k  entries  each.  The  matrix  must  be  a  listing  of  the  num- 
bers from  0  to  2^^  —  1  in  binary  notation.  The  other  n  —  k  columns  of 
the  original  matrix  considered  are  linear  combinations  of  the  columns  of 
this  matrix.  This  completes  the  proof  of  Theorem  3  and  Proposition  4. 

2.6  PROOF    OF   THEOREM    4 

To  prove  Theorem  4  we  first  note  that  the  parity  check  sequence  of 
the  product  of  two  elements  of  Bn  is  the  mod  2  sum  of  their  separate 


A    CLASS    OF    BINARY    SIGNALING   ALPHABETS  225 

parity  check  sequences.  It  follows  then  that  all  elements  in  a  given  coset 
have  the  same  parity  check  sequence.  For,  let  the  coset  be  Si ,  SiA2 , 
SiAz ,  ■  ■  •  SiA^  .  Since  the  elements  I,  A^ ,  A3,  •  •  •  ,  A^  all  have  parity 
check  sequence  00  •  •  •  0,  all  elements  of  the  coset  have  parity  check 
R(Si). 

In  the  array  (4)  there  are  2"  cosets.  We  observe  that  there  are  2"~* 
elements  of  Bn  that  have  zeros  in  their  first  k  places.  These  elements 
have  parity  check  symbols  identical  with  the  last  n  —  k  places  of  their 
symbols.  These  elements  therefore  give  rise  to  2"~  different  parity  check 
symbols.  The  elements  must  be  distributed  one  per  coset.  This  proves 
Theorem  4. 

2.7    PROOF   OF   PROPOSITION    5 

If 

n  ^  2"-'  - 

we  can  explicity  exhibit  group  alphabets  having  the  property  mentioned 
in  the  paragraph  preceding  Proposition  5.  The  notation  of  the  demon- 
stration is  cumbersome,  but  the  idea  is  relatively  simple. 

We  shall  use  the  notation  of  paragraph  2.1  for  elements  of  Bn  ,  i.e., 
an  element  of  Bn  will  be  given  by  a  list  of  integers  that  specify  what 
places  of  the  sequence  for  the  element  contain  ones.  It  will  be  convenient 
furthermore  to  designate  the  first  k  places  of  a  sequence  by  the  integers 
1,  2,  3,  •  •  •  ,  k  and  the  remaining  n  —  k  places  by  the  "integers"  1',  2', 
3',  •  •  •  ,  r,  where  (  =  n  —  k.  For  example,  if  n  =  8,  /c  =  5,  we  have 

10111010^  13452' 
10000100^  11' 
00000101  ^  1'3' 

Consider  the  group  generated  by  the  elements  1',  2',  3',  •  •  •  ,  (' ,  i.e. 
the  2'  elements  /,  1',  2',  ■■■,(',  1'2',  1'3',  •  •  •  ,  1'2'3'  ■■■('.  Suppose 
these  elements  listed  according  to  decreasing  weight  (say  in  decreasing 
order  when  regarded  as  numbers  in  the  decimal  system)  and  numbered 
consecutively.  Let  Bt  be  the  zth  element  in  the  list.  Example:  if  (  ^  3, 
Ih  =  1'2'3',  B2  =  2'3',  B,  =  1'3',  B,  =  1'2',  B,  -  3',  B,  =  2',  B,  -  1'. 

Consider  now  the  (n,  /^-alphabet  whose  generators  are 

ISi  ,  2B,  ,W,,  ■■•  ,  kBk 
We  assert  that  if 


22G  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 

>r%  n—k 
..  —  2        - 


this  alphabet  is  as  good  as  any  other  alphabet  of  2   letters  and  n  places. 

In  the  first  place,  we  observe  that  every  letter  of  this  (n,  A-)-alphabet 
(except  /)  has  unprimed  numbers  in  its  symbols.  It  follows  that  each  of 
the  2'  letters  /,  1',  2',  •  ■  •  ,  (',  V2',  ■■■  ,  V2'  ■■■  ('  occurs  in  a  different 
coset  of  the  given  (n,  A-)-alphabet.  For,  if  two  of  these  letters  appeared 
in  the  same  coset,  their  product  (which  contains  only  primed  numbers) 
would  have  to  be  a  letter  of  the  (n,  k)  alphabet.  This  is  impossible  since 
every  letter  of  the  (/i,  A)  alphabet  has  unprimed  numbers  in  its  symbol. 
Since  there  are  precisely  2  cosets  we  can  designate  a  coset  by  the  single 
element  of  the  list  Bi ,  Bi  ,  ■  •  ■  ,  B-ii  =  I  which  appears  in  the  coset. 

We  next  observe  that  the  condition 

71    ^    2  — 

guarantees  that  J5a+i  is  of  weight  3  or  less.  For,  the  given  condition  is 
equivalent  to 

'-■-©-o-o-e 

We  treat  several  cases  depending  on  the  weight  of  Bu+i . 

If  Bk+\  is  of  weight  3,  we  note  that  for  i  =  1,2,  •  •  •  ,  A-,  the  coset  con- 
taining Bi  also  contains  an  element  of  weight  one,  namely  the  element 
i  obtained  as  the  product  of  Bi  with  the  letter  iBi  of  the  given  (n,  A;)- 
alphabet.  Of  the  remaining  (2    —  A')  5's,  one  is  of  weight  zero,  C  are  of 

weight  one,  f     j  are  of  weight  2  and  the  remaining  are  of  weight  3.  We 

have,  then  an  =  1,  ai  =  f  +  A-  =  n.  Now  every  B  of  weight  4  occurs  in' 
the  list  of  generators  \Bi  ,  2B-2  ,  •  •  •  ,  kBk  .  It  follows  that  on  multi- 
plying this  list  of  generators  by  any  B  of  weight  3,  at  least  one  element 
of  weight  two  will  result.  (E.g.,  (l'2'3')(il'2'3'40  =  j4')  Thus  every 
coset  with  a  B  of  weight  2  or  3  contains  an  element  of  weight  2  and 
a2  =  2     —  ao  —  cn]  . 

The  argument  in  case  Bk+i  is  of  weight  two  or  one  is  similar. 

2.8   MODULAR    REPRESENTATIONS    OF    C„ 

In  order  to  explain  one  of  the  methods  used  to  obtain  the  best  (//,  A)- 
alphabets  listed  in  Tal)les  II  and  III,  it  is  necessary  to  digress  here  lo 
present  additional  theory. 


I 


A    CLASS    OF    BINARY   SINGALING   ALPHABETS 


227 


It  has  been  remarked  that  every  (n,  /v)-alphabet  is  isomorphic  with 
Ck  .  Let  us  suppose  the  elements  of  Ci,  hsted  in  a  column  starting  with  / 
and  proceeding  in  order  /,  1,  2,  3,  •  •  •  ,  /.',  12,  13,  ■••,(/.•—   1)/,-,  123, 

,   123  •  •  •  k.  The  elements  of  a  given  (n,  A-)-alphabet  can  be 

paired  off  with  these  abstract  elements  so  as  to  preserve  group  multipli- 
cation. This  can  be  done  in  many  different  ways.  The  result  is  a  matrix 
with  elements  zero  and  one  with  7i  columns  and  2  rows,  these  latter 
being  labelled  by  the  symbols  /,  1,2,  •  •  •  etc.  What  can  be  said  about 
the  columns  of  this  matrix?  How  many  different  columns  are  possible 
when  all  (n,  A)-alphabets  and  all  methods  of  establishing  isomorphism 
with  Ck  are  considered? 

In  a  given  column,  once  the  entries  in  rows  1,2,  •  •  •  ,  /,•  are  known,  the 
entire  column  is  determined  by  the  group  property.  There  are  therefore 
only  2  possible  different  columns  for  such  a  matrix.  A  table  showing 
these  2  possible  columns  of  zeros  and  ones  will  be  called  a  modular  repre- 
senfafion  table  for  Ck  ■  An  example  of  such  a  table  is  shown  for  /,•  =  4  in 
Table  VI. 

It  is  clear  that  the  colunuis  of  a  modular  representation  table  can  also 
be  labelled  by  the  elements  of  Ck  ,  and  that  group  multiplication  of  these 
column  labels  is  isomorphic  with  mod  2  addition  of  the  columns.  The 
table  is  a  symmetric  matrix.  The  element  with  row  label  A  and  column 
label  B  is  one  if  the  symbols  A  and  B  have  an  odd  number  of  different 
numerals  in  common  and  is  zero  otherwise. 

Every  (n,  /c)-alphabet  can  be  made  from  a  modular  representation 
table  by  choosing  w  columns  of  the  table  (with  possible  repetitions)  at 
least  k  of  which  form  an  independent  set. 


Table  VI  —  Modular  Representation  Table  for  Group  C4 

I    12    3    4    12   13   14   23   24   34   123    124   134    234   1234 

I 

1 

2 

3 

4 

12 

13 

14 

23 

24 

34 

123 

124 

134 

234 

1234 


0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

n 

0 

1 

0 

0 

0 

1 

1 

1 

1 

0 

0 

1 

0 

0 

1 

0 

1 

1 

0 

0 

0 

1 

0 

0 

0 

0 

1 

0 

0 

0 

0 

1 

0 

1 

1 

1 

0 

1 

1 

0 

0 

0 

1 

0 

0 

0 

1 

0 

1 

0 

1 

1 

1 

0 

0 

1 

0 

0 

1 

1 

0 

0 

0 

0 

0 

1 

1 

0 

1 

0 

1 

0 

0 

0 

1 

0 

1 

1 

1 

0 

0 

0 

0 

0 

1 

1 

0 

1 

1 

0 

0 

1 

1 

1 

() 

0 

1 

0 

1 

0 

1 

1 

0 

1 

0 

0 

1 

1 

0 

1 

0 

1 

1 

1 

0 

0 

1 

0 

0 

1 

1 

1 

1 

1 

0 

1 

0 

1 

1 

1 

1 

0 

0 

0 

0 

1 

u 

228  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,   JANUARY    1956 

We  henceforth  exclude  consideration  of  the  column  /  of  a  modular 
representation  table.  Its  inckision  in  an  (n,  /v)-alphabet  is  clearly  a  waste 
of  1  binary  digit. 

It  is  easy  to  show  that  every  column  of  a  modular  representation  table 
for  Ch  contains  exactly  2  "  ones.  Since  an  (n,  /v)-alphabet  is  made  from 
n  such  columns  the  alphabet  contains  a  total  of  n2 '~  ones  and  we  have 

Proposition  6.  The  weights  of  an  (n,  /c)-alphabet  form  a  partition  of 
n2''~^  into  2*  —  1  non-zero  parts,  each  part  being  an  integer  from  the  set 
1,2,  ■■■  ,n. 
The  identity  element  always  has  weight  zero,  of  course. 

It  is  readily  established  that  the  product  of  two  elements  of  even 
weight  is  again  an  element  of  even  weight  as  is  the  product  of  two  ele- 
ments of  odd  weight.  The  product  of  an  element  of  even  weight  with  an 
element  of  odd  weight  yields  an  element  of  odd  weight. 

The  elements  of  even  weight  of  an  (n,  A;) -alphabet  form  a  subgroup 
and  the  preceding  argument  shows  that  this  subgroup  must  be  of  order 
2*"  or  2*""^  If  the  group  of  even  elements  is  of  order  2''~\  then  the  collec- 
tion of  even  elements  is  a  possible  (n,  k  —  l)-alphabet.  This  (n,  k  —  1) 
alphabet  may,  however,  contain  the  column  /  of  the  modular  represen- 
tation table  of  Ck-i  ■  We  therefore  have 

Proposition  7.  The  partition  of  Proposition  6  must  be  either  into 
2^  —  1  even  parts  or  else  into  2  "  odd  parts  and  2^—1  even  parts. 
In  the  latter  case,  the  even  parts  form  a  partition  of  a2  ""  where  a  is 
some  integer  of  the  set  k  —  I,  k,  ■  •  •  ,  n  and  each  of  the  parts  is  an  in- 
teger from  the  set  1,  2,  •  •  •  ,  n. 

2.9   THE    CHARACTERS    OF    Ck 

Let  us  replace  the  elements  of  Bn  (each  of  which  is  a  sequence  of  zeros 
and  ones)  by  sequences  of  4-1 's  and  —  I's  by  means  of  the  following 
substitution 

The  multiplicative  properties  of  elements  of  Bn  can  be  preserved  iti  this 
new  notation  if  we  define  the  product  of  two  4-1,-1  symbols  to  be  the 
symbol  whose  tth  component  is  the  ordinary  product  of  the  ?'th  compo- 
nents of  the  two  factors.  For  example,  1011  and  01 10  become  respectively 
-11  -1  -1  and  1  -1  -11.  We  have 

(-11  -1  -1)(1  -1  -11)  =  (-1  -11  -1) 


1 

0 

0 

0 

0 

1 

0 

0 

0 

0 

-1 

0 

0 

0 

0 

-1 

A    CLASS    OF   BINARY   SIGNALING   ALPHABETS  229 

corresponding  to  the  fact  that 

(1011)  (0110)  =  (1101) 

If  the  +1,-1  symbols  are  regarded  as  shorthand  for  diagonal  matrices, 
so  that  for  example 


-11  -1  -1 


then  group  multiplication  corresponds  to  matrix  multiplication. 

(While  much  of  what  follows  here  can  be  established  in  an  elementary 
way  for  the  simple  group  at  hand,  it  is  convenient  to  fall  back  upon  the 
established  general  theory  of  group  representations  for  several  proposi- 
tions. 

The  substitution  (13)  converts  a  modular  representation  table  (col- 
umn /  included)  into  a  square  array  of  +l's  and  —  I's.  Each  column  (or 
row)  of  this  array  is  clearly  an  irreducible  representation  of  Ck  ■  Since  Ck 
is  Abelian  it  has  precisely  2  irreducible  representations  each  of  degree 
one.  These  are  furnished  by  the  converted  modular  table.  This  table  also 
furnishes  then  the  characters  of  the  irreducible  representations  of  Ck 
and  we  refer  to  it  henceforth  as  a  character  table. 

Let  x"(^)  be  the  entry  of  the  character  table  in  the  row  labelled  A  and 
column  labelled  a.  The  orthogonality  relationship  for  characters  gives 


E  x'{A)/{A)  =  2'8., 


ACCk 


Z  x%A)x"(B)  =  2'b 

<xCCk 


AB 


where  8  is  the  usual  Kronecker  symbol.  In  particular 

E  xiA)x\A)  =    Z  AA)  =  0,        ^^I 

ACCk  ACCk 

Since  each  x  (A)  is  +1  or  —  1,  these  must  occur  in  eciual  numbers  in  any 
column  ^  9^  I.  This  implies  that  each  column  except  /  of  the  modular 
representation  table  contains  2  ~  ones,  a  fact  used  earlier. 

Every  matrix  representation  of  Ck  can  be  reduced  to  its  irreducible 
components.  If  the  trace  of  the  matrix  representing  the  element  A  in  an 
arbitrary  matrix  representation  of  Ck  is  x{A),  then  this  representation 
contains  the  irreducible  representation  having  label  ^  in  the  character 
table  dp  times  where 


230  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


(h  =  ^.    E    x{A)AA)  (14) 


2^- 


A  C  Ck 


Every  (n,  A)-alphabet  furnishes  iis  with  a  matrix  representation  of  Ck 
by  means  of  (13)  and  the  procedure  outUned  below  (13).  The  trace  xi^.) 
of  the  matrix  representing  the  element  A  of  C\  is  related  to  the  weight 
of  the  letter  by 

x(A)  =  n  -  2w(A)  (15) 

Equations  (14)  and  (15)  permit  us  to  compute  from  the  weights  of  an 
(u,  /,)-alphabet  what  irreducible  representations  are  present  in  the  alpha- 
bet and  how  many  times  each  is  contained.  It  is  assumed  here  that  the 
given  alphabet  has  been  made  isomorphic  to  Ck  and  that  the  weights  are 
labelled  by  elements  of  Ck  ■ 

Consider  the  converse  problem.  Given  a  set  of  mmibers  ivi  ,  Wn  ,  •  ■  ■  , 
W'lk  that  satisfy  Propositions  6  and  7.  From  these  we  can  compute 
cjuantities  %/  =  n  —  2wi  as  in  (15).  It  is  clear  that  the  given  ty's  will 
constitute  the  weights  of  an  (/t,  A)-alphabet  if  and  only  if  the  2^  x»  can 
be  labelled  with  elements  of  (\  so  that  the  2  sums  (14)  {fi  ranges  over 
all  elements  of  Ck)  are  non-negative  integers.  The  integers  d^  tell  what 
representations  to  choose  to  construct  an  in,  A)-alphabet  with  the  given 
weights  Wi  . 

2.10  CONSTRUCTION  OF  BEST  ALPHABETS 

A  great  many  different  techniques  were  used  to  construct  the  group 
alphabets  listed  in  Tables  II  and  III  and  to  show  that  for  each  n  and  k 
there  are  no  group  alphabets  with  smaller  probability  of  error.  Space 
prohibits  the  exhibition  of  proofs  for  all  the  alphabets  listed.  We  content 
ourseh'es  here  with  a  sample  argument  and  treat  the  case  n  =  10,  k  = 
4  in  detail. 

According  to  (2)  there  are  A^(10,  4)  =  53,743,987  different  (10,  4)- 
alphabets.  We  now  show  that  none  is  better  than  the  one  given  in  Table 
III.  The  letters  of  this  alphabet  and  weights  of  the  letters  are 

1  0 

167  8  10  5 

2  6  7  9  10  5 

3  5  6  8  9  10  6 

4  5  7  8  9  10  6 
1289  4 
13579  5 


A    CLASS   OF    BINARY   SIGNALING   ALPHABETS 


231 


14569 
23578 
24568 
3  4  6  7 
12  3  5  7  9 
12  4  5  7  10 

1  3  4  8  10 

2  3  4  9  10 

12  3  4  6  7  8  9 


5 
5 
5 
4 
6 
6 
5 
5 
8 


The  notation  is  that  of  Section  2.1.  By  actually  forming  the  standard 
array  of  this  alphabet,  it  is  verified  that 


ao  =1,         Oil  =  10, 


«2 


39, 


a:i 


14. 


Table  II  shows  (  .->  )  =  ^5,  whereas  a-z  =  39,  so  the  given  alphabet 

does  not  correct  all  possible  double  errors.  In  the  standard  array  for  the 
alphabet,  39  coset  leaders  are  of  weight  2.  Of  these  39  cosets,  33  have 
only  one  element  of  weight  2;  the  remaining  6  cosets  each  contain  two 
elements  of  weight  2.  This  is  due  to  the  two  elements  of  weight  4  in  the 
given  group,  namely  1289  and  3467.  A  portion  of  the  standard  array 
that  demonstrates  these  points  is 


1289 


3467 


12 

89 

• 

18 

29 

• 

19 

28 

. 

34 

67 

36 

47 

37 

46 

] 

• 

In  order  to  have  a  smaller  probability  of  error  than  the  exhibited 
alphabet,  it  is  necessary  that  a  (10,  4)-alphabet  have  an  a^  >  39.  We 
proceed  to  show  that  this  is  impossible  by  consideration  of  the  weights 
of  the  letters  of  possible  (10,  4)-alphabets. 

We  first  show  that  every  (10,  4)-alphabet  must  have  at  least  one  ele- 
ment (other  than  the  identity,  /)  of  weight  less  than  5.  By  Propositions 
•  ')  and  7,  Section 2.8,  the  weights  must  form  a  partition  of  10-8  =  80  into 
1 5  positive  parts.  If  the  weights  are  all  even,  at  least  two  must  be  less 
than 6  since  14-6  =  84  >  80.  If  eight  of  the  weights  are  odd,  we  see  from 
8-5  +  7-()  =  82  >  80  that  at  least  one  weight  must  be  less  than  5. 


232  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

An  alphabet  with  one  or  more  elements  of  weight  1  must  have  an 
«2  ^  36,  for  there  are  nine  elements  of  weight  2  which  cannot  possibly 
be  coset  leaders.  To  see  this,  suppose  (without  loss  of  generality)  that 
the  alphabet  contains  the  letter  1.  The  elements  12,  13,  14,  •  •  •  1  10  can- 
not possibly  be  coset  leaders  since  the  product  of  any  one  of  them  with 
the  letter  1  yields  an  element  of  weight  1 . 

An  alphabet  with  one  or  more  elements  of  weight  2  must  have  an 
ai  S  37.  Suppose  for  example,  the  alphabet  contained  the  letter  12. 
Then  13  and  23  must  be  in  the  same  coset,  14  and  24  must  be  in  the 
same  coset,  ■  •  •  ,  1  10  and  2  10  must  be  in  the  same  coset.  There  are  at 
least  eight  elements  of  weight  two  which  are  not  coset  leaders. 

Each  element  of  weight  3  in  the  alphabet  prevents  three  elements  of 
weight  2  from  being  coset  leaders.  For  example,  if  the  alphabet  contains 
123,  then  12,  13,  and  23  cannot  be  coset  leaders.  We  say  that  the  three 
elements  of  weight  2  are  "blocked"  by  the  letter  of  weight  3.  Suppose  an 
alphabet  contains  at  least  three  letters  of  weight  three.  There  are  several 
cases:  (A)  if  three  letters  have  no  numerals  in  common,  e.g.,  123,  456, 
789,  then  nine  distinct  elements  of  weight  2  are  blocked  and  a-2  S  36; 
(B)  if  no  two  of  the  letters  have  more  than  a  single  numeral  in  common, 
e.g.,  123,  345,  789,  then  again  nine  elements  of  weight  2  are  blocked  and 
a-2  ^  36;  and  (C)  if  two  of  the  letters  of  weight  3  have  two  numerals  in 
common,  e.g.,  123,  234,  then  their  product  is  a  letter  of  weight  2  and  l)y 
the  preceding  paragraph  ao  ^  37.  If  an  alphabet  contains  exactly  two 
elements  of  weight  3  and  no  elements  of  weight  2,  the  elements  of  weight 

3  block  six  elements  of  weight  2  and  0:2  ^  39. 
The  preceding  argument  shows  that  to  be  better  than  the  exhibited 

alphabet  a  (10,  4)-alphabet  with  letters  of  weight  3  must  have  just  one 
such  letter.  A  similar  argument  (omitted  here)  shows  that  to  be  better 
than  the  exhibited  alphabet,  a  (10,  4)-alphabet  cannot  contain  more 
than  one  element  of  weight  4.  Furthermore,  it  is  easily  seen  that  an 
alphabet  containing  one  element  of  weight  3  and  one  element  of  weight 

4  must  have  an  ao  ^  39. 
The  only  new  contenders  for  best   (10,  4)-alphabet  are,  therefore, 

alphabets  with  a  single  letter  other  than  /  of  weight  less  than  5,  and  this 
letter  must  have  weight  3  or  4.  Application  of  Propositions  6  and  7  show 
that  the  only  possible  weights  for  alphabets  of  this  sort  are:  35  6   and 

5  46'  where  5'  means  seven  letters  of  weight  5,  etc.  We  next  show  that 
there  do  not  exist  (10,  4)-alphabets  having  these  weights. 

Consider  first  the  suggested  alphabet  with  weights  35  6'.  As  explained 
in  Section  2.9,  from  such  an  alphabet  we  can  construct  a  matrix  repre- 
sentation of  ('4  having  the  character  x(/)  =   10,  one  matrix  of  trace  4, 


A   CLASS    OF    BINARY   SIGNALING   ALPHABETS  233 

seven  of  trace  0  and  seven  of  trace  —2.  The  latter  seven  matrices  cor- 
respond to  elements  of  even  weight  and  together  with  /  must  represent 
a  subgroup  of  order  8.  We  associate  them  with  the  subgroup  generated 
by  the  elements  2,  3,  and  4.  We  have  therefore 

x(/)  =  10,        x(2)  =  x(3)  =  x(4)  =  x(23) 

=  x(24)  =  x(34)  =  x(234)  =  -2. 

Examination  of  the  symmetries  involved  shows  that  it  doesn't  matter 
how  the  remaining  Xi  ai"e  associated  with  the  remaining  group  elements. 
We  take,  for  example 

x(l)  =  4,        x(12)  =  x(13)  =  x(14)  =  x(123) 

=  x(124)  =  x(134)  =  x(1234)  =  0. 

Now  form  the  sum  shown  in  equation  (14)  with  /3  =  1234  (i.e.,  with  the 
character  x^"  obtained  from  column  1234  of  the  Table  VI  by  means 
of  substitution  (13).  There  results  c?i234  =  V-i  which  is  impossible.  There- 
fore there  does  not  exist  a  (10,  4) -alphabet  with  weights  35  6  . 

The  weights  5  46  correspond  to  a  representation  of  d  with  character 
x(/)  =  10,  0^,  2,  (  — 2)^  We  take  the  subgroup  of  elements  of  even  weight 
to  be  generated  by  2,  3,  and  4.  Except  for  the  identity,  it  is  clearly  im- 
material to  w^hich  of  these  elements  we  assign  the  character  2.  We  make 
the  following  assignment:  x(/)  =  10,  x(2)  =  2,  x(3)  =  x(4)  =  x(23)  = 
x(24)  =  x(34)  =  x(234)  =  -2,  x(l)  =  x(12)  =  x(13)  =  x(14)  = 
x(123)  =  x(124)  =  x(134)  =  x(1234)  =  0.  The  use  of  equation  (14) 
shows  that  ^2  =  \'2  which  is  impossible. 

It  follows  that  of  the  53,743,987  (10,  4)-alphabets,  none  is  better  than 
the  one  listed  on  Table  III. 

Not  all  the  entries  of  Table  III  were  established  in  the  manner  just 
demonstrated  for  the  (10,  4)-alphabet.  In  many  cases  the  search  for  a 
l)est  alphabet  was  narrowed  down  to  a  few  alphabets  by  simple  argu- 
ments. The  standard  arrays  for  the  alphabets  were  constructed  and  the 
best  alphabet  chosen.  For  large  n  the  labor  in  making  such  a  table  can 
be  considerable  and  the  operations  involved  are  highly  liable  to  error 
when  performed  by  hand. 

I  am  deeply  indebted  to  V.  M.  Wolontis  who  programmed  the  IBM 
CPC  computer  to  determine  the  a's  of  a  given  alphabet  and  who  pa- 
tiently ran  off  many  such  alphabets  in  course  of  the  construction  of 
Tables  II  and  III.  I  am  also  indebted  to  Mrs.  D.  R.  Fursdon  who  eval- 
uated many  of  the  smaller  alphabets  by  hand. 


234  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 


REFERENCES 

1.  R.  W.  Hamming,  B.S.T.J.,  29,  i)p.  147-160,  1950. 

2.  I.  S.  Reed,  Transactions  of  tlie  Piofossional  (iroup  on  Information  Tlieorv, 
^  PGIT-4,  PI).  3S-49,  1954. 

3.  See  section  7  of  R .  W.  Hamniinji's  paper,  loc.  cit. 

4.  I.R.E.    Convention    Record,    I'art   4,    pp.    37-45,    1955    National    Convention, 

March,  1955. 

5.  C.  E.  Shannon,  B.S.T.J.,  27,  pp.  379-423  and  pp.  623-656,  1948. 

6.  Birkhoff  and  MacLane,  A  Snrvey  of  Modern  Algebra,  Macmillan  Co.,  New 

York,  1941 .  Van  der  Waerden,  Alodern  Algebra,  Ungar  Co.,  New  York,  1953. 
Miller,  Bliclifeldt,  and  Dickson,  Finite  Groups,  Stechert,  New  York,  1938. 

7.  This  theorem  has  been  previously  noted  in  the  literature  by  Kiyasu-Zen'iti, 

Research  and  Development  Data  No.  4,  Ele.  Comm.  Lai).,  Nippon  Tele. 
Corp.  Tokyo,  Aug.,  1953. 

8.  F.  D.  Murnaghan,  Theory  of  Group  Representations,  Johns  Hopkins  Press, 

Baltimore,  1938.  E.  Wigner,  Gruppentheorie,  Edwards  Brothers,  Ann  Arbor, 
Michigan,  1944. 


I 


Bell  System  Technical  Papers  Not 
Published  in  This  Journal 

Allen,  L.  J.,  see  Fewer,  D.  R. 

Alllson,  H.  W.,  see  Moore,  G.  E. 

Baker,  W.  0.,  see  Winslow,  F.  H. 

Barstow,  J.  M.^ 

Color  TV       How  it  Works,  I.R.E.  Student  Quarterly,  2,  pp.  11-16, 
Sept.,  1955. 

Basseches,  H.^  and  ^McLean,  D.  A. 

Gassing  of  Liquid  Dielectrics  Under  Electrical  Stress,  Ind.  c^-  Engg. 
Chem.,  47,  pp.  1782-1794,  Sept.,  1955. 

Beck,  A.  C} 

Measurement  Techniques  for  Multimode  Waveguides,  Proc.  I.R.E., 
MRI,  4,  pp.  325-6,  Oct.  1,  1955. 

Becker,  J.  A.^ 

The  Life  History  of  Adsorbed  Atoms,  Ions,  and  Molecules,  N.  Y. 
Acad.  Sci.  Ann.,  58,  pp.  723-740,  Sept.  15,  1955. 

Hlackwell,  J.  H.,  see  Fewer,  D.  R. 
BooRSE,  H.  A.,  see  Smith,  B. 

HozoRTii,  R.  M.,'  Getlin,  B.  B.,'  Galt,  J.  K.,'  Merritt,  F.  R.,'  and- 

^'ager,  W.  a.' 
Frequency  Dependence  of  Magnetocrystalline  Anisotropy,  Letter  to 
the  Editor,  Phys.  Rev.,  99,  p.  1898,  Sept.  15,  1955. 


1.  Bell  Telephone  Laboratories,  Inc. 

235 


236  THE   BELL   SYSTEM   TECHXICAL   JOURNAL,    JANUARY    1956 

BozoRTH.  R.  M.\  TiLDEX,  E.  F..'  and  Williams,  A.  j/ 
Anisotropy  and  Magnetostriction  of  Some  Ferrites,  Phys.  Rev.,  99, 
pp.  17S8-1798,  Sept.  15,  1955. 

Bridgers,  H.  E.,^  and  Kolb,  E.  D.^ 

Rate-Grown  Germanium  Crystals  for  High-Frequency  Transistors, 
Letter  to  the  Editor,  J.  Appl.  Phys.,  26,  pp.  1188-1189,  Sept.,  1955.   j 

BULLIXGTOX,  K.^ 

Characteristics  of  Beyond-the-Horizon  Radio   Transmission,  Pioc. 
I.R.E.,  43,  pp.  1175-1180,  Oct.,  1955. 

BULLIXGTOX,  K.^  IXKSTER,  W.  J.,^  and  DVRKEE,  A.  L.^ 

Results  of  Propagation  Tests  at  505  Mc  and  4,090  Mc  on  Beyond- 
Horizon  Paths,  Proc.  I.R.E.,  43,  pp.  1306-1316,  Oct.,  1955. 

Calbick,  C.  J.' 

Surface  Studies  with  the  Electron  Microscope,  X.  Y.  Acad.  Sci.  Ann., 
58,  pp.  873-892,  Sept.  15,  1955. 

Cass,  R.  S.,  see  Fewer,  D.  R. 

DuRKEE,  A.  L.,  see  Bullington,  K. 

Fewer,  D.  R..'  Blackwell.  J.  H..'  Allex.  L.  J..^  and  Cass,  R.  S." 
Audio-Frequency  Circuit  Model  of  the  1-Dimensional  Schroedinger 
Equation  and  Its  Sources  of  Error,  Canadian  J.  of  Pins.,  33,  pp.  483- 
491,  Aug.,  1955. 

Francois,  E.  E.,  see  Law,  J.  T. 

Davis,  J.  L.,  see  Suhl,  H. 

Galt,  J.  K.,  see  Bozorth,  R.  "SI.,  and  Yager,  W.  A. 

Garn,  p.  D.,'  and  Hallixe,  Mrs.  E.  W.' 

Polarographic  Determination  of  Phthalic  and  Anhydride  Alkyd  Res- 
ins, Anal  Cliem.,  27,  pp.  15()3-15G5,  Oct.,  1955. 

1.  Bell  Telephone  Laboratories,  Inc. 

4.  University  of  Western  Ontario,  London,  Canada 

5.  Bell  Telephone  Company  of  Canada,  Montreal 


TECHNICAL    PAPERS  237 

Getlin,  B.  B.,  see  Bozorth,  R.  M. 

GlANOLA,  V.  F} 

Application  of  the  Wiedemann  Effect  to  the  Magnetostrictive  Coupling 
of  Crossed  Coils,  J.  Appl.  Phys.,  26,  pp.  1152-1157,  Sept.,  1955. 

Goss,  A.  J.,  see  Hassion,  F.  X. 

Green,  E.  I.^ 
The  Story  of  0,  American  Scientist,  43:  pp.  584-594,  Oct.,  1955. 

Halline,  Mrs.  E.  W.,  see  Garn,  P.  D. 

Harrower,  G.  A.^ 
Measurement  of  Electron  Energies  by  Deflection  in  a  Uniform  Electric 
Field,  Rev.  Sci.  Instr.,  26,  pp.  850-854,  Sept.,  1955. 

Hassion,  F.  X.,^  Goss,  A.  .1.,^  and  Trumbore,  F.  A.^ 
The  Germanium-Silicon  Phase  Diagram,  J.  Phys.  Chem.,  59,  p.  1118, 
Oct.,  1955. 

Hassion,  F.  X.,^  Thurmond,  C.  D.,^  and  Trumbore,  F.  A.^ 

On  the  Melting  Point  of  Germanium,  J.  Phys.  Chem.,  59,  p.  1076, 
Oct.,  1955. 

Hines,  I\I.  E.,'  Hoffman,  G.  W.,'  and  Saloom,  J.  A.^ 

Positive-Ion  Drainage  in  Magnetically  Focused  Electron  Beams,  J. 
Appl.  Phys.,  26,  pp.  1157-1162,  Sept.,  1955. 

Hoffman,  G.  W.,  see  Hines,  M.  E. 

Inkster,  W.  J.,  see  Bullington,  K. 

Kelly,  M.  J.' 
Training  Programs  of  Industry  for  Graduate  Engineers,  Elec.  Engg., 
74,  pp.  866-869,  Oct.,  1955. 

KoLB,  E.  D.,  see  Bridgers,  H.  E. 
1.  Bell  Telephone  Laboratories,  Inc. 


1 


238  THE    BELL    SYSTEM   TECHXICAL   JOURXAL,    JANUARY    1 9 of) 

Law,  J.  T./  and  Francois,  E.  E.' 

Adsorption  of  Gasses  and  Vapors  on  Germanium,  X.  Y.  Acad.  Sci. 
Ann.,  58,  pp.  925-936,  Sept.  15,  1955. 

LovELL,  Miss  L.  C,  see  Pfann,  W.  G. 

Matreyek,  W.,  see  Winslow,  F.  H. 

McLean,  D.  A.,  see  Basseches,  H. 

Merritt,  F.  R.,  see  Bozorth,  R.  M.,  and  Yager,  W.  A. 

Meyer,  F.  T.' 

An  Improved  Detached-Contact  Type  of  Schematic  Circuit  Drawing, 
A.LE.E.  Commun.  ct  Electronics,  20,  pp.  505-513,  Sept.,  1955. 

Miller,  B.  T.' 

Telephone  Merchandising,  Telephony,   149,  pp.   116-117,  Oct.  22, 
1955. 

Miller,  S.  L.^ 

Avalanche  Breakdown  in  Germanium,  Phys.  Rev.,  99,  pp.  1234-1241, 
Aug.  15,  1955. 

Moore,  G.  E.,^  and  Allison,  H.  W.^ 
Adsorption   of  Strontium  and  of  Barium   on   Tungsten,   J.   Chem. 
Phys.,  23,  pp.  1609-1621,  Sept.,  1955. 

Neisser,  W.  R.,^ 

Liquid  Nitrogen  Coal  Traps,  Rev.  Sci.  Instr.,  26,  p.  305,  Mar.,  1955. 

Ostergren,  C.  N." 

Some  Observations  on  Liberahzed  Tax  Depreciation,  Telephony,  149, 
pp.  16-23-37,  Oct.  1,  1955. 

Ostergren,  G.  N. 

Depreciation  and  the  New  Law,  Telephony,  149,  pp.  96-100-104-108, ; 
Oct.  22,  1955.  I 

Rape,  N.  R.,  see  Winslow,  F.  H. 

1.  Bell  Telephone  Laboratories,  Inc. 

2.  American  Telephone  and  Telegraph  Co. 


\\ 


technical  papers  239 

Pedekskn,  L. 
Aluminum  Die  Castings  for  Carrier  Telephone  Systems,  A.I.E.E. 
Commun.  &  Electronics,  20,  pp.  434-439,  Sept.,  1955. 

Peters,  H.^ 
Hard  Rubber,  Tnd.  and  Engg.  Chem.,  Part  II,  pp.  2220-2222,  Sept. 
20,  1955. 

Pfann,  w.  c;.' 

Temperature-Gradient  Zone-Melting,  J.  Metals,  7,  p.  961,  Sept.,  1955. 

Pfann,  W.  G.,'  and  Lovell,  Miss  L.  C.^ 
Dislocation  Densities  in  Intersecting  Lineage  Boundaries  in  Ger- 
manium, Letter  to  the  Editor,  Acta.  Met.,  3,  pp.  512-513,  Sept.,  1955. 

Pierce,  J.  P.' 
Orbital  Radio  Relays,  Jet  Propulsion,  25,  pp.  153-157,  Apr.,  1955. 

Poole,  K.  M.' 
Emission  from  Hollow  Cathodes,  J.  Appl.  Phys.,  26,  pp.  1176-1179, 
Sept.,  1955. 

Saloom,  J.  A.,  see  Hines,  M.  E. 

Slighter,  W.  P.^ 
Proton  Magnetic  Resonance  in  Polyamides,  J.  Appl.  Phys.,  26,  pp., 
1099-1103,  Sept.,  1955. 

Smith,  B./  and  Boorse,  H.  A. 
Helium  II  Film  Transport.  II.  The  Role  of  Surface  Finish,  Phys.  Rev. 
99,  pp.  346-357,  July  15,  1955. 

Smith,  B.,^  and  Boorse,  H.  A. 
Helium  II  Film  Transport.  IV.  The  Role  of  Temperature,  Phys.  Rev., 
99,  pp.  367-370,  July  lo,  1955. 

SuHL,  H.,^  Van  Uitert,  L.  G.,^  and  Davis,  J.  L.^ 
Ferromagnetic  Resonance  in  Magnesium-Manganese  Aluminum  Fer- 
rite  Between  160  and  1900  Mc,  Letter  to  the  Editor,  J.  Appl.  Phys., 
26,  pp.  1181-1182,  Sept.,  1955. 

1.  Bell  Telephone  Laboratories,  Inc. 

6.  Columbia  University,  New  York  City 


240  THE    EELL   SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1956 

Thurmond,  C.  D.,  see  Hassion,  F.  X. 

TiDD,  W.  H/  I 

Demonstration  of  Bandwidth  Capabilities  of  Beyond -Horizon  Tropo- 
spheric  Radio  Propagation,  Proc.  I.R.E.,  43,  pp.  1297-1299,  Oct.,  1955. 

Tien,  P.  K.,'  and  Walker,  L.  R.' 
Large  Signal  Theory  of  Traveling -Wave  Amplifiers,  Proc.  I.R.E.,  43, 
p.  1007,  Aug.,  1955. 

TiLDEN,  E.  F.,  see  Bozorth,  R.  M. 

Trumbore,  F.  a.,  see  Hassion,  F.  X. 

IThlir,  a.,  Jr.^ 

Micromachining  with  Virtual  Electrodes,  Rev.  Sci.,  Instr.,  26,  pp. 
965-968,  Oct.,  1955. 

Ulrich,  W.,  see  Yokelson,  B.  J. 

Van  Uitert,  L.  G.,  see  Siihl,  H. 

Walker,  L.  R.,  see  Tien,  P.  K. 

Weibel,  E.  S.' 
Vowel  Synthesis  by  Means  of  Resonant  Circuits,  J.  Acous.  Soc,  27, 
pp.  858-865,  Sept.,  1955. 

Williams,  A.  J.,  see  Bozorth,  R.  M. 

WiNSLow,  F.  H.,'  Baker,  W.  O.,^  and  Yager,  W.  A.^ 

Odd  Electrons  in  Polymer  Molecules,  Am.  Chem.  Soc,  77,  pp.  4751- 
4756,  Sept.  20,  1955. 

WiNSLow,  F.  II.,'  Baker,  W.  O.,'  Rape,  N.  R.'  and  Matreyek,  W.' 
Formation  and  Properties  of  Polymer  Carbon,  J.  Polymer  Science,  16, 
p.  101,  Apr.,  1955. 

Yager,  W.  A.,  sec  Bozorth,  R.  M. 
1.  Bell  Tc;l(;i)li()ne  liaboratorics,  Inc. 


TECHNICAL   PAPERS  241 

Yagkr,  W.  a./  Galt,  J.  K/  and  Merritt,  F.  R.' 
Ferromagnetic  Resonance  in  Two-Nickel-Iron  Ferrites,  Phys.  Rev., 
99,  pp.  1203-1209,  Aug.  15,  1955. 

YoKELSON,  B.  J.,^  and  Ulrich,  W.^ 

Engineering  Multistage  Diode  Logic  Circuits,  A.I.E.E.  Commun.  & 
Electronics,  20,  pp.  -466-475,  Sept.,  1955. 

1.  Bell  Telephone  Laboratories,  Inc. 


Recent  Monographs  of  Bell  System  Technical 
Papers  Not  Published  in  This  Journal* 

Arnold,  W.  O.,  and  Hoefle,  R.  R. 

A  System  Plan  for  Air  Traffic  Control,  ]\Ionograph  2483. 

Beck,  A.  C. 

Measurement  Techniques  for  Multimode  Waveguides,  ]\Ioiiograph 
2421. 

Becker,  J.  A.,  and  Brandes,  R.  G. 

Adsorption  of  Oxygen  on  Tungsten  as  Revealed  in  Field  Emission 
Microscope,  Alonogiaph  24U3. 

Boyle,  W.  S.,  see  Germer,  L.  H. 

Brandes,  R.  G.,  see  Becker,  J.  A. 

Brattain,  W.  H.,  see  Garrett,  C.  G.  B. 

Garrett,  C.  G.  B.,  and  Brattain,  W.  H. 

Physical  Theory  of  Semiconductor  Surfaces,  Monograph  2453. 

Gerner,  L.  H.,  Boyle,  W.  S.,  and  Kisliuk,  P. 

Discharges  at  Electrical  Contacts  —  II,  Monograph  2499. 

Hoefle,  R.  R.,  see  Arnold,  W.  0. 

KisLiuK,  P.,  see  Germer,  L.  H. 

Linvill,  J.  G. 

Nonsaturating  Pulse  Circuits  Using  Two  Junction  Transistors,  Mono- 
graph 2-17.").  I 

*  Copies  of  these  monographs  may  1)0  ()l)l;tin(Ml  on  request  to  the  Pul)licat ion 
Department,  Hell  Telephone  Laboratories,  Iiie.,  463  West  Street,  New  York  14, 
N.  Y.  The  numbers  of  the  monographs  should  be  given  in  all  requests. 

242 


MONOGRAPHS  243 

Mason,  W.  P. 

Relaxations  in  the  Attenuation  of  Single  Crystal  Lead,  Monograph 
2454. 

Mkykr,  F.  T. 
An  Improved  Detached-Contact-Type  of  Schematic  Circuit  Drawing, 
Monograph  2456. 

VoGEL,  F.  L.,  Jr. 

Dislocations  in  Low-Angle  Boundaries  in  Germanium,  Monograph 
2455. 

Walker,  T..  R. 

Generalizations  of  Brillouin  Flow,  Monograph  2432. 

Warner,  A.  W. 

Frequency  Aging  of  High -Frequency  Plated  Crystal  Units,  Monograph 
2474. 

Weibel,  E.  S. 

On  Webster's  Horn  Equation,  Monograph  2450. 


Contributors  to  This  Issue 

A.  C.  Beck,  E.E.,  Rensselaer  Polytechnic  Institute,  1927;  Instructor, 
Rensselaer  Polytechnic  Institute,  1927-1928;  Bell  Telephone  Labora- 
tories, 1928  -.  With  the  Radio  Research  Department  he  was  engaged 
in  the  development  and  design  of  short-wave  and  microwave  antennas. 
During  World  War  II  he  was  chiefly  concerned  with  radar  antennas  and 
associated  waveguide  structures  and  components.  For  several  years 
after  the  war  he  worked  on  development  of  microwave  radio  repeater 
systems.  Later  he  worked  on  microwave  transmission  developments 
for  broadband  communication.  Recently  he  has  concentrated  on  further 
developments  in  the  field  of  broadband  communication  using  circular 
waveguides  and  associated  test  equipment. 

J.  S.  Cook,  B.E.E.,  and  M.S.,  Ohio  State  University,  1952;  Bell 
Telephone  Laboratories,  1952  -.  Mr.  Cook  is  a  member  of  the  Research 
in  High-Frequency  and  Electronics  Department  at  Murray  Hill  and 
has  been  engaged  principally  in  research  on  the  traveling- wave  tube. 
Mr.  Cook  is  a  member  of  the  Institute  of  Radio  Engineers  and  belongs 
to  the  Professional  Group  on  Electron  Devices. 

0.  E.  DeLange,  B.S.  University  of  Utah,  1930;  M.A.  Columbia  Uni- 
versity, 1937;  Bell  Telephone  Laboratories,  1930  — .  His  early  work  was 
principally  on  the  development  of  high-frequency  transmitters  and  re- 
ceivers. Later  he  worked  on  frequency  modulation  and  during  World 
War  II  was  concerned  with  the  development  of  radar.  Since  that  time 
he  has  been  involved  in  research  using  broadband  systems  including 
microwa^'e  and  baseband.  Mr.  DeLange  is  a  member  of  the  Institute 
of  Radio  Engineers. 

R.  KoMPFNER,  Engineering  Degree,  Technische  Hochschule,  Vienna, 
1933;  Ph.D.,  Oxford,  1951;  Bell  Telephone  Laboratories,  1951  -.  Be- 
tween 1941-1950  he  did  work  for  the  British  Admiralty  at  Birmingham 
University  and  Oxford  University  in  the  Royal  Naval  Scientific  Service. 
He  invented  the  traveling-wave  tube  and  for  this  achievement  Dr. 
Kompfner  i-eceived  the  1955  Duddcll  Medal,  bestowed  by  the  Physical 
Society  of  England.  In  the  Laboratoi'ies'  Research  in  High  Frequency 

244 


CONTRIBUTORS   TO   THIS   ISSUE  245 

and  Electronics  Department,  he  has  continued  his  research  on  vacuum 
tubes,  particularly  those  used  in  the  microwave  region.  He  is  a  Fellow 
of  the  Institute  of  Radio  Engineers  and  of  the  Physical  Society  in 
London. 

Charles  A.  Lee,  B.E.E.,  Rensselaer  Polytechnic  Institute,  1943; 
Ph.D.,  Columbia  University,  1953;  Bell  Telephone  Laboratories,  1953-. 
When  Mr.  Lee  joined  the  Laboratories  he  became  engaged  in  research 
concerning  solid  state  devices.  In  particular  he  has  been  developing 
techniques  to  extend  the  frequency  of  operation  of  transistors  into  the 
microwave  range,  including  work  on  the  diffused  base  transistor.  During 
World  War  II,  as  a  member  of  the  United  States  Signal  Corps,  he  was 
concerned  with  the  determination  and  detection  of  enemy  counter- 
measures  in  connection  with  the  use  of  proximity  fuses  by  the  Allies. 
He  is  a  member  of  the  American  Physical  Society  and  the  American 
Institute  of  Physics.  He  is  also  a  member  of  Sigma  Xi,  Tau  Beta  Pi 
and  Eta  Kappa  Nu. 

John  R.  Pierce,  B.S.,  M.S.  and  Ph.D.,  California  Institute  of  Tech- 
nology 1933,  1934  and  1936;  Bell  Telephone  Laboratories,  1936-.  Ap- 
pointed Director  of  Research  —  Electrical  Communications  in  August, 
1955.  Dr.  Pierce  has  specialized  in  Development  of  Electron  Tubes  and 
Microwave  Research  since  joining  the  Laboratories.  During  World  War 
li  II  he  concentrated  on  the  development  of  electronic  devices  for  the 
[I  Armed  Forces.  Since  the  war  he  has  done  research  leading  to  the  develop- 
;j  ment  of  the  beam  traveling- wave  tube  for  which  he  was  awarded  the 
h  1947  Morris  Liebmann  Memorial  Prize  of  the  Institute  of  Radio  Engi- 
[li  neers.  Dr.  Pierce  is  author  of  two  books:  Theory  and  Design  of  Electron 
Ij  Beams,  published  in  second  edition  last  year,  and  Traveling  Wave  Tubes 
il  (1950).  He  was  voted  the  ''Outstanding  Young  Electrical  Engineer  of 
[|  1942"  by  Eta  Kappa  Nu.  Fellow  of  the  American  Physical  Society  and 
J  the  I.R.E.  Member  of  the  National  Academy  of  Sciences,  the  A.I.E.E., 
I  Tau  Beta  Pi,  Sigma  Xi,  Eta  Kappa  Nu,  the  British  Interplanetary  So- 
il ciety,  and  the  Newcomen  Society  of  North  America. 

C.  F.  QuATE,  B.S.,  University  of  Utah  1944;  Ph.D.,  Stanford  Uni- 
i  versity  1950;  Bell  Laboratories  1950-.  Dr.  Quate  has  been  engaged  in 
rj  research  on  electron  dynamics  —  the  study  of  vacuum  tubes  in  the 
;|  microwave  frequency  range.  He  is  a  member  of  I.R.E. 

I      David  Slepian,  University  of  Michigan,  1941-1943;  M.A.  and  Ph.D., 
li  Harvard  LTniversity,  1946-1949;  Bell  Telephone  Laboratories,  1950-.  Dr. 


24G  THE    BELL    SYSTEM   TECHNICAL   JOURNAL,    JANUARY    1950 

Slepian  has  been  engaged  in  mathematical  research  in  communication 
theory,  switching  theory  and  theory  of  noise.  Parker  Fellow  in  physics. 
Harvard  University  1949-50.  Member  of  I.R.E,,  American  Mathemati- 
cal Society,  the  American  Association  for  the  Advancement  of  Science 
and  Sigma  Xi. 

Milton  Sobel,  B.S.,  City  College  of  New  York,  1940;  M.A.,  1946  and 
Ph.D.,  1951,  Columbia  University;  U.  S.  Census  Bureau,  Statistician, 
1940-41;  U.  S.  Army  War  College,  Statistician,  1942-44;  Cohunbia  Uni- 
versity, Department  of  Mathematics,  Assistant,  1946-48  and  Research 
Associate  1948-50;  Wayne  University,  Assistant  Professor  of  Mathe- 
matics, 1950-52;  Columbia  University,  Department  of  Mathematical 
Statistics,  Visiting  Lecturer,  1952;  Cornell  University,  fundamental  re- 
search in  mathematical  statistics,  1952-54;  Bell  Telephone  Laboratories, 
1954-.  Dr.  Sobel  is  engaged  in  fundamental  research  on  life  testing 
reliability  problems  with  special  application  to  transistors  and  is  a  con- 
sultant on  many  Laboratories  projects.  Member  of  Institute  of  Mathe- 
matical Statistics,  American  Statistical  Association  and  Sigma  Xi. 

Morris  Tanenbaum,  A.B.,  Johns  Hopkins  University,  1949;  M.A., 
Princeton  University,  1950;  Ph.D.  Princeton  University,  1952;  Bell 
Telephone  Laboratories,  1952-,  Dr.  Tanenbaum  has  been  concerned 
with  the  chemistry  and  semiconducting  properties  of  intermetallic  com- 
pounds. At  present  he  is  exploring  the  semiconducting  properties  of 
silicon  and  the  feasibility  of  silicon  semiconductor  devices.  Dr.  Tanen- 
baum is  a  member  of  the  American  Chemical  Society  and  American 
Physical  Society.  He  is  also  a  member  of  Phi  Lambda  LTpsilon,  Phi  Beta 
Kappa  and  Sigma  Xi. 

Donald  E.  Thomas,  B.S.  in  E.E.,  Pennsylvania  State  College,  1929; 
M.A.,  Columbia  University,  1932;  Bell  Telephone  Laboratories,  1929- 
1942,  1946-.  His  first  assignment  at  the  Laboratories  was  in  submarine 
cable  development.  Just  prior  to  World  War  II  he  became  engaged  in 
the  development  of  sea  and  airborne  radar  and  continued  in  this  work  I 
until  he  left  for  military  duty  in  1942.  During  World  War  II  he  was  made ' 
a  member  of  the  Joint  and  Combined  Chiefs  of  Staff  Committees  on 
Radio  C-ountermeasures.  Later  he  was  a  civilian  memlior  of  the  Depart-' 
ment  of  Defense's  Research  and  Development  Board  Panel  on  Electronic 
Countermeasures.  Upon  rejoining  the  Laboratories  in  1946,  Mr.  Thomas 
was  active  in  the  development  and  installation  of  the  first  deep  sea  re- 
peatered  submarine  telephone  cable,  hctwcen  Key  West  and  Havana,' 


COXTIUBUTOKS   TO   THIS   ISSUE  247 

which  went  into  service  in  1950.  Later  he  was  engaged  in  the  develop- 
ment of  transistor  devices  and  circuits  for  special  applications.  At  the 
present  time  he  is  working  on  the  evaluation  and  feasibility  studies  of 
new  types  of  semiconductors  devices.  He  is  a  senior  member  of  the  I.R.E. 
and  a  member  of  Tau  Beta  Pi  and  Phi  Kappa  Phi. 

Laurence  R.  Walker,  B.Sc.  and  Ph.D.,  McGill  University,  1935 
and  1939;  LTniversity  of  California  1939-41;  Radiation  Laboratory, 
Massachusetts  Institute  of  Technology,  1941-45;  Bell  Telephone  La- 
boratories, 1945-.  Dr.  Walker  has  been  primarily  engaged  in  the  develop- 
ment of  microwave  oscillators  and  amplifiers.  At  present  he  is  a  member 
of  a  physical  research  group  concerned  with  the  applied  physics  of  solids. 
Fellow  of  the  American  Physical  Society. 


IHE      BELL      SYSTEM 

Jechnical  journal 

VOTED    TO    THE    SC  I  E  N  T  I  FIC^^^    AND    ENGINEERING 
PECTS    OF    ELECTRICAL    COMMUNICATION 


LUME  XXXV  MARCH    1956  NUMBER  2 


An  Experimental  Remote  Controlled  Line  Concentrator  \.f^  y 

A^E.  JOEL,  JR.  249 

Transistor  Circuits  for  Analog  and  Digital  Systems 

F.  H.  BLECHER   295 

Electrolytic  Shaping  of  Germanium  and  Silicon  a.  uhlir,  jr.  333 

A  Large  Signal  Theory  of  Traveling-Wave  Amplifiers        p.  k.  tibn  349 

A  Detailed  Analysis  of  Beam  Formation  with  Electron  Guns  of  the 
Pierce  Type    w.  e.  danielson,  j.  l.  rosenfeld  and  j.  a.  saloom  375 

Theories  for  Toll  Traffic  Engineering  in  the  U.S.A.    r.  i,  Wilkinson  421 

Crosstalk  on  Open -Wire  Lines 

W,  C,  BABCOCK,  ESTHER  RENTROP  AND  C.  S.  THAELER  515 


Bell  System  Technical  Papers  Not  Published  in  This  Journal  519 

Recent  Bell  System  Monographs  527 

Contributors  to  This  Issue  531 


COPYRIGHT  1956  AMERICAN  TELEPHONE  AND  TELEGRAPH  COMPANY 


THE  BELL  SYSTEM  TECHNICAL  JOURNAL 


ADVISORY    BOARD 

F.  K.  K  A  P  P  E  L,  President  Western  Electric  Company 

M.  J.  KELLY,  President,  Bell  Telephone  Laboratories 

E.  J.  McNEELY,  Executive  Vice  President,  American 
Telephone  and  Telegraph  Company 

EDITORIAL    COMMITTEE 

B.  MCMILLAN,  Chairman 

A.  J.  BUSCH  H.  R.  HUNTLEY 

A.   C.   DICKIBSON  F.   R.   LACK 

R.   L.   DIETZOLD  J.   R.   PIERCE 

K.  E.  GOULD  H.   V.   SCHMIDT 

E.   I.   GREEN  C.   ESCHOOLEY 

R.   K.  HON  AM  AN  G.  N.  THAYER 

ED ITORI AL    STAFF 

J.  D.  TEBO,  Editor 

M.  E.  s  T  R  I  E  B  Y,  Managing  Editor 

R.  L.  SHEPHERD,  Production  Editor 


THE  BELL  SYSTEM  TECHNICAL  JOURNAL  is  published  six  times 
a  year  by  the  American  Telephone  and  Telegraph  Company,  195  Broadway, 
New  York  7,  N.  Y.  Qeo  F.  Craig,  President;  S.  Whitney  Landon,  Secretary; 
John  J.  Scanlon,  Treasurer.  Subscriptions  are  accepted  at  $3.00  per  year. 
Single  copies  are  75  cents  each.  The  foreign  postage  is  65  cents  per  year  or  11 
cents  per  copy.  Printed  in  U.  S.  A. 


THE   BELL  SYSTEM 

TECHNICAL  JOURNAL 

VOLUME  XXXV  MARCH   1956  number  2 

Copyright  1958,  American  Telephone  and  Telegraph  Company 

An  Experimental  Remote  Controlled 
Line  Concentrator 

By.  A.  E.  JOEL,  JR. 

(Manuscript  received  June  30,  1955) 

Concentration,  which  is  the  process  of  connecting  a  number  of  telephone 
lines  to  a  smaller  number  of  switching  paths,  has  always  been  a  funda?nental 
function  in  switching  systems.  By  performing  this  function  remotely  from 
the  central  office,  a  new  balance  between  outside  plant  and  switching  costs 
may  be  obtained  which  shows  promise  of  providing  service  more  economi- 
cally in  some  situations. 

The  broad  concept  of  remote  line  concentrators  is  not  new.  However,  its 
solution  with  the  new  devices  and  techniques  now  available  has  made  the 
possibilities  of  decentralization  of  the  means  for  switching  telephone  con- 
nections very  promising. 

Three  models  of  an  experimental  equipment  have  been  designed  and  con- 
structed for  service.  The  models  have  included  equipment  to  enable  the  evalua- 
tion of  new  procedures  required  by  the  introduction  of  remote  line  concentra- 
tors into  the  telephone  plant.  The  paper  discusses  the  philosophy,  devices, 
and  techniques. 

CONTENTS 

1 .  Introduction 250 

2.  Objectives 251 

3.  New  Devices  Emploj^ed 252 

4.  New  Techniques  Emploved 254 

5.  Switching  Plan ". 257 

249 


250  THE   BELL   SYSTEM   TECHNICAL  JOURNAL,    MARCH    1956 

6.  Basic  Circuits 261 

a.  Diode  Gates 261 

b.  Transistor  Bistable  Circuit 262 

c.  Transistor  Pulse  Amplifier 263 

d.  Transistor  Ring  Counter 264 

e.  Crosspoint  Operating  Circuit 266 

f .  Crosspoint  Relay  Circuit 267 

g.  Pulse  Signalling  Circuit 268 

h.  Power  Supply 269 

7.  Concentrator  Operation 270 

a.Line  Scanning 270 

b.  Line  Selection 272 

c.  Crosspoint  Operation  and  Check 273 

8.  Central  Office  Circuits 274 

a.  Scanner  Pulse  Generator 279 

b.  Originating  Call  Detection  and  Line  Number  Registration 280 

c.  Line  Selection 282 

d.  Trunk  Selection  and  Identification 284 

9.  Field  Trials 286 

10.  Miscellaneous  Features  of  Trial  Equipment 287 

a.  Traffic  Recorder,        b.  Line  Condition  Tester 288 

c.  Simulator,        d.  Service  Observing 290 

e.  Service  Denial,        f .   Pulse  Display  Circuit 291 

1.   INTRODUCTION 

The  equipment  which  provides  for  the  switching  of  telephone  connec- 
tions has  ahvays  been  located  in  what  have  been  commonly  called  "cen- 
tral offices".  These  offices  provide  a  means  for  the  accumulation  of  all 
switching  equipment  required  to  handle  the  telephone  needs  of  a  com- 
munity or  a  section  of  the  community.  The  telephone  building  in  which 
one  or  more  central  offices  are  located  is  sometimes  referred  to  as  the 
"wire  center"  because,  like  the  spokes  of  a  wheel,  the  wires  which  serve 
local  telephones  radiate  in  all  directions  to  the  telephones  of  the 
community. 

A  new  development,  made  possible  largely  by  the  application  of  de- 
vices and  techniques  new  to  the  telephone  switching  field,  has  recently 
been  tried  out  in  the  telephone  plant  and  promises  to  change  much  of  . 
the  present  conception  of  "central"  offices  and  "wire"  centers.  It  is 
known  as  a  "line  concentrator"  and  provides  a  means  for  reducing  the 
amount  of  outside  plant  cables,  poles,  etc.,  serving  a  telephone  central 
office  by  dispersing  the  switching  equipment  in  the  outside  plant.  It  is 
not  a  new  concept  to  reduce  outside  plant  by  bringing  the  switching 
equipment  closer  to  the  telephone  customer  but  the  technical  difficulties 
of  maintaining  complex  switching  equipment  and  the  cost  of  controlling" 
such  equipment  at  a  distance  have  in  the  past  been  formidable  obstacles 
to  the  development  of  line  concentrators.  With  the  invention  of  low 
power,  small-sized,  long-life  devices  such  as  transistors,  gas  tubes,  and 
sealed  relays,  and  their  application  to  line  concentrators,  and  with  the 
development  of  new  local  switching  systems  with  greater  flcxibilit}',  it 
has  been  possible  to  make  the  progress  described  herein. 


REMOTE   CONTROLLED   LINE   CONCENTRATOR  251 

2.  OBJECTIVES 

Within  the  telephone  offices  the  first  switching  equipment  through 

which  dial  lines  originate  calls  concentrates  the  traffic  to  the  remaining 

equipment  which  is  engineered  to  handle  the  peak  busy  hour  load  with 

the  appropriate  grade  of  service.^  This  concentration  stage  is  different  for 

different  switching  systems.  In  the  step-by-step  system^  it  is  the  line 

'  finder,  and  in  the  crossbar  systems  it  is  the  primary  line  switch.^  Pro- 

1  posals  for  the  application  of  remote  line  concentrators  in  the  step-by- 

i  step  system  date  back  over  50  years/  Continuing  studies  over  the  years 

have  not  indicated  that  any  appreciable  savings  could  be  realized  when 

such  equipment  is  used  within  the  local  area  served  by  a  switching  center. 

When  telephone  customers  move  from  one  location  to  another  within 

a  local  service  area,  it  is  desirable  to  retain  the  same  telephone  numbers. 

The  step-by-step  switching  system  in  general  is  a  unilateral  arrangement 

where  each  line  has  two  appearances  in  the  switching  equipment,  one 

for  originating  call  concentration  (the  line  finder)  and  one  for  selection 

of  the  line  on  terminating  calls  (the  connector) .  The  connector  fixes  the 

line  number  and  telephone  numbers  cannot  be  readily  reassigned  when 

moving  these  switching  stages  to  out-of-office  locations. 

Common-control  systems^  have  been  designed  with  flexibility  so  that 
the  line  number  assignments  on  the  switching  equipment  are  independ- 
ent of  the  telephone  numbers.  Furthermore,  the  first  switching  stage 
in  the  office  is  bilateral,  handling  both  originating  and  terminating  calls 
through  the  same  facilities.  The  most  recent  common-control  switching 
system  in  use  in  the  Bell  System,  the  No.  5  crossbar,^  has  the  further 
advantage  of  universal  control  circuitry  for  handling  originating  and 
terminating  calls  through  the  line  switches.  For  these  reasons,  the  No. 
5  crossbar  system  was  chosen  for  the  first  attempt  to  employ  new  tech- 
niques of  achieving  an  economical  remote  line  concentrator. 

A  number  of  assumptions  were  made  in  setting  the  design  require- 
ments. Some  of  these  are  influenced  by  the  characteristics  of  the  No.  5 
crossbar  system.  These  assumptions  are  as  follows: 

1.  No  change  in  customer  station  apparatus.  Standard  dial  telephones 
to  be  used  with  present  impedance  levels,  transmission  characteristics, 
dial  pulsing,  party  identification,  superimposed  ac-dc  ringing,^  and  sig- 
naling and  talking  ranges. 

2.  Individual  and  two-party  (full  or  semi-selective  ringing)  stations 
to  be  served  but  not  coin  or  PBX  lines. 

3.  Low  cost  could  best  be  obtained  by  minimizing  the  per  line 
equipment  in  the  central  office.  AMA^  charging  facilities  could  be  used 
but  to  avoid  per  station  equipment  in  the  central  office  no  message  reg- 
ister operation  would  be  provided. 


252  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 

4.  Each  concentrator  would  serve  up  to  50  lines  with  the  central  office 
control  circuits  common  to  a  number  of  concentrators.  (Experimental 
equipment  described  herein  was  designed  for  60  lines  to  provide  addi- 
tional facilities  for  field  trial  purposes.)  No  extensive  change  would  be 
made  in  central  office  equipment  not  associated  with  the  line  switches 
nor  should  concentrator  design  decrease  call  carrying  capacities  of  exist- 
ing central  office  equipment. 

5.  To  provide  data  to  evaluate  service  performance,  automatic  traffic 
recording  facilities  to  be  integrated  with  the  design. 

6.  Remote  equipment  designed  for  pole  or  wall  mounting  as  an  addi- 
tion to  existing  outside  plant.  Therefore,  terminal  distribution  facilities 
would  not  be  provided  in  the  same  cabinet. 

7.  Power  to  be  supplied  from  the  central  office  to  insure  continuity 
of  telephone  service  in  the  event  of  a  local  power  failure. 

8.  Concentrators  to  operate  over  existing  types  of  exchange  area  fa- 
cilities without  change  and  with  no  decrease  in  station  to  central  office 
service  range. 

9.  Maintenance  effort  to  be  facilitated  by  plug-in  unit  design  using 
the  most  reliable  devices  obtainable. 

3.    NEW   DEVICES   EMPLOYED 


»! 


I 


Numerous  products  of  research  and  development  were  available  for 
this  new  approach.  Only  those  chosen  will  be  described. 

For  the  switching  or  "crosspoint"  element  itself,  the  sealed  reed  switch 
was  chosen,  primarily  because  of  its  imperviousness  to  dirt.*  A  short  coil 
magnet  with  magnetic  shield  for  increasing  sensitivity  of  the  reed 
switches  were  used  to  form  a  relay  per  crosspoint  (see  Fig.  1). 

A  number  of  switching  applications^ '^^  for  crosspoint  control  using 
small  gas  diodes  have  been  proposed  by  E.  Bruce  of  our  Switching  Re- 
search Department.  They  are  particularly  advantageous  when  used  in 
an  "end  marking"  arrangement  with  reed  relay  crosspoints.  Also,  these 
diodes  have  long  life  and  are  low  in  cost.  One  gas  diode  is  employed  for 
operating  each  crosspoint  (see  Fig.  6).  Its  breakdown  voltage  is  125v  ± 
lOv,  A  different  tube  is  used  in  the  concentrator  for  detecting  marking 
potentials  when  termination  occurs.  Its  breakdown  potential  is  lOOv  ± 
lOv.  One  of  these  tubes  is  used  on  each  connection. 

Signaling  between  the  remote  concentrator  and  the  central  office  con- 
trol circuits  is  performed  on  a  sequential  basis  with  pulses  indicative  of 
the  various  line  conditions  being  transmitted  at  a  500  cycle  rate.  This 
frequency  encounters  relatively  low  attenuation  on  existing  exchange 
area  wire  facilities  and  j^et  is  high  enough  to  transmit  and  receive  in- 
formation at  a  rate  which  will  not  decrease  call  carrjdng  capacitj^  of  the 


REMOTE  CONTROLLED  LINE  CONCENTRATOR 


253 


Fig.  1  —  Reed  switch  relay. 

central  office  equipment.  To  accomplish  this  signaling  and  to  process  the 
information  economically  transistors  appear  most  promising. 

Germanium  alloy  junction  transistors  were  chosen  because  of  their 
;  improved  characteristics,  reliability,  low  power  requirements,  and  mar- 
gins, particularly  when  used  to  operate  with  relays.^^  Both  N-P-N  and 
P-N-P  transistors  are  used.  High  temperature  characteristics  are  par- 
ticularly important  because  of  the  ambient  conditions  which  obtain  on 
pole  mounted  equipment.  As  the  trials  of  this  equipment  have  progressed, 


254  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


Table  I— Transistor  Characteristics 


Code  No. 

Type  and  Filling 

Alpha 

Max.  Ico  at  28V 
and  65°C 

Emitter  Zener 
Voltage  at  20^=1 

M1868 
M1887 

p-n-p  Oxygen 
n-p-n  Vacuum 

0.9-1.0 
0.5-  .75 

150  Ma 
100  Ma 

>735 
>735 

considerable  progress  has  been  made  in  improving  transistors  of  thi.s 
type.  Table  I  summarizes  the  characteristics  of  these  transistors. 

For  directing  and  analyzing  the  pulses,  the  control  employs  semicon- 
ductor diode  gate  circuits."  The  semiconductor  diodes  used  in  these 
circuits  are  of  the  silicon  alloy  junction  type,^^  Except  for  a  few  diode.s 
operating  in  the  gas  tube  circuits  most  diodes  have  a  breakdown  voltage 
requirement  of  27v,  a  minimum  forward  current  of  15  ma  at  2v  and  a 
maximum  reverse  current  at  22v  of  2  X  10^^  amp. 

4.  new  techniques  employed 

The  concentrator  represents  the  first  field  application  in  Bell  System 
telephone  switching  systems  which  departs  from  current  practices  and 
techniques.  These  include: 


Fig.  2  —  Transistor   packages,    (a)   Diode    unit,    (b)   Transistor  counter,    (c) 
Transistor  amplifiers  and  bi-stable  circuits,  (d)  Five  trunk  unit. 


REMOTE    CONTROLLED    LINE    CONCENTRATOR  255 

1.  High  speed  pulsing  (500  pulses  per  second)  of  information  between 
switching  units. 

2.  The  use  of  plug-in  packages  employing  printed  wiring  and  encap- 
sulation. (Fig.  2  shows  a  representative  group  of  these  units.) 

3.  Line  scanning  for  supervision  with  a  passive  line  circuit.  In  present 
systems  each  line  is  equipped  with  a  relay  circuit  for  detecting  call  orig- 
inations (service  requests)  and  another  relay  (or  switch  magnet)  for 
indicating  the  busy  or  idle  condition  of  the  line,  as  shown  in  Fig.  3(a). 
The  line  concentrator  utilizes  a  circuit  consisting  of  resistors  and  semi- 
conductor diodes  in  pulse  gates  to  provide  these  same  indications.  This 
circuit  is  shown  in  Fig.  3(b).  Its  operation  is  described  later.  The  pulses 
for  each  line  appear  at  a  different  time  with  respect  to  one  another. 
These  pulses  are  said  to  represent  "time  slots."  Thus  a  different  line  is 
examined  each  .002  second  for  a  total  cycle  time  (for  60  lines)  of  .120 
second.  This  process  is  known  as  "line  scanning"  and  the  portion  of  the 
circuit  which  produces  these  pulses  is  known  as  the  scanner.  Each  of  the 
circuits  perform  the  same  functions,  viz.,  to  indicate  to  the  central  office 
equipment  when  the  customer  originates  a  call  and  for  terminating  calls 
to  indicate  if  the  line  is  busy. 

4.  The  lines  are  divided  for  control  and  identification  purposes  into 
twelve  groups  of  five  lines  each.  Each  group  of  five  lines  has  a  different 
pattern  of  access  to  the  trunks  which  connect  to  the  central  office.  The 
ten  trunks  to  the  central  office  are  divided  into  two  groups  as  shown  in 
Fig.  4.  One  trunk  group,  called  the  random  access  group,  is  arranged  in 
a  random  multiple  fashion,  so  that  each  of  these  trunks  is  available  to 
approximately  one-half  of  the  lines.  The  other  group,  consisting  of  two 
trunks,  is  available  to  all  lines  and  is  therefore  called  the  full  access 
group.  The  control  circuitry  is  arranged  to  first  select  a  trunk  of  the 
random  access  group  which  is  idle  and  available  to  the  particular  line  to 
which  a  connection  is  to  be  made.  If  all  of  the  trunks  of  this  random  ac- 
cess group  are  busy  to  a  line  to  which  a  connection  is  desired,  an  attempt 
is  then  made  to  select  a  trunk  of  the  full  access  group.  The  preference 
order  for  selecting  cross-points  in  the  random  access  group  is  different 
for  each  line  group,  as  shown  in  the  table  on  Fig.  4.  By  this  means,  each 
trunk  serves  a  number  of  lines  on  a  different  priority  basis.  Random  ac- 
cess is  used  to  reduce  by  40  per  cent  the  number  of  individual  reed  relay 
crosspoints  which  would  otherwise  be  needed  to  maintain  the  quality 
of  service  desired,  as  indicated  by  a  theory  presented  some  years  ago.^^ 

5.  Built-in  magnetic  tape  means  for  recording  usage  data  and  making 
call  delay  measurements.  The  gathering  of  this  data  is  greatly  facilitated 
by  the  line  scanning  technique. 


256 


THE   BELL    SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


CROSSBAR  CROSSPOINT 

OR 

SWITCH    CONTACTS 


-^ 


TO  LINE 


-^ 


TO  OTHER 

CENTRAL  OFFICE 

EQUIPMENT 


9  9 


r 


^ 


LR 


■^f- 


CO 


c 


HI 


"H 


1_ 


(a) 


■:l 


LINE    BUSY 


SERVICE 
REQUEST 


I  +  5V 


CROSSPOINT 


■^ 


TO  LINE 


-^ 


TO 

CENTRAL 

OFFICE 


^4- 


-X 


-16V 


-16  VOLTS -NORMAL 
(RECEIVER  ON   HOOK) 

-3   VOLTS -AWAITING  SERVICE 
(RECEIVER  OFF   HOOK) 

-16  VOLTS -CROSSPOINT  CLOSED 
(RECEIVER  OFF  HOOK) 


S\. 


-¥^ 


-16  V 


^ 


LINE  BUSY 


+  15  VOLT 
TIME  SLOT  PULSE 
FROM    SCANNER 


GATE 


SERVICE 
REQUEST 


Fig.  3  —  (a)  Relay  line  circuit,  (b)  Passive  line  circuit. 


REMOTE    CONTROLLED    LINE    CONCENTRATOR 


257 


5.    SWITCHING   PLAN 

The  plan  for  serving  lines  directly  terminating  in  a  No.  5  Crossbar  office 
is  shown  in  Fig.  5(a).  Each  line  has  access  through  a  primary  line  switch 
to  10  line  links.  The  line  links  couple  the  primary  and  secondary  switches 
together  so  that  each  line  has  access  to  all  of  the  100  junctors  to  the  trunk 
link  switching  stage.  Each  primary  line  switch  group  accommodates 
from  19  to  59  lines  (one  line  terminal  being  reserved  for  no-test  calls). 
A  line  link  frame  contains  10  groups  of  primary  line  switches.^* 
.  The  remote  concentrator  plan  merely  extends  these  line  links  as  trunks 
to  the  remote  location.  However,  an  extra  crossbar  switching  stage  is 
introduced  in  the  central  office  to  connect  the  links  to  the  secondary  line 
switches  with  the  concentrator  trunks  as  shown  in  Fig.  5(b).  Since  each 
line  does  not  have  full  access  to  the  trunks,  the  path  chosen  by  the  marker 
to  complete  calls  through  the  trunk  link  frame  may  then  be  independent 
of  the  selection  of  a  concentrator  trunk  with  access  to  the  line.  This 
arrangement  minimizes  call  blocking,  simplifies  the  selection  of  a  matched 
path  by  the  marker,  and  the  additional  crossbar  switch  hold  magnet 
serves  also  as  a  supervisory  relay  to  initiate  the  transmission  of  disconnect 
signals  over  the  trunk. 

In  addition  to  the  10  concentrator  trunks  used  for  talking  paths,  2 
additional  cable  pairs  are  provided  from  each  concentrator  to  the  central 
office  for  signaling  and  power  supply  purposes.  The  use  of  these  two  pairs 
of  control  conductors  is  described  in  detail  in  Section  6g. 

The  concentrator  acts  as  a  slave  unit  under  complete  control  of  the 
central  office.  The  line  busy  and  service  request  signals  originate  at  the 


LINE 


60    LINES 

I 


o.-»-o 

0 

5 

9 

7            '^ 

/      ^ 

/      ■v 

/ 

p,          ■^ 

i' 

\. 

^ 

V 

•y 

\ 

/        s 

f 

\ 

^ 

\ 

,•      \ 

'^ 

^ 

1              > 

f 

<  > 

1      2      3 


5     6 


8      9     10     11 


Fig  4.  —  Concentrator  trunk 
to  line  crosspoint  pattern  and 
preference  order 


CONCENTRATOR 
TRUNKS 


9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

8 

8 

8 

8 

8 

8 

8 

8 

8 

8 

8 

8 

6 

0 

5 

4 

7 

5 

3 

1 

4 

7 

2 

1 

7 

3 

1 

5 

2 

0 

6 

4 

6 

5 

0 

3 

1 

7 

2 

3 

6 

2 

4 

0 

0 

6 

3 

5 

0 

4 

6 

2 

3 

7 

1 

6 

2 

4 

1 

7 

1 


5      6 


8 


VERTICAL    GROUPS    OF    FIVE    LINES    EACH " 

ORDER    OF    PREFERENCE 


GAS    TUBE    REED -RELAY 
CROSS    POINTS 


10     11 


258 


THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


49 
LINES 


TEN    GROUPS 
OF    LINES 


49 

LINES  I 


CENTRAL    OFFICE 
LINE    LINK  FRAME 


LINE  SW 


1 

CON- 
NECTOR 

1 
1 

1 

— 

-- 1 

1 

CON- 
NECTOR 

I 

TO    MARKER 


TRUNK 
LINK 
FRAME 


Fig.  5(a)  —  No.  5  crossbar  system  subscriber  lines  connected  to  line  link  frame. 


60 

LINES 


60 
LINES 


60 
0      <? 


10 


CONTROL 


CE^4TRAL  OFFICE 


TEN    CONCENTRATOR 
,  TRUNKS 


I 

JL 


TWO   CONTROL    PAIRS 


60 

0     0 


10 


TEN    POLE- 
MOUNTED 
^CONCENTRATOR 
UNITS   AT 
DIFFERENT 
LOCATIONS 


CONTROL 


TEN  CONCENTRATOR 
TRUNKS 


TWO   CONTROL  PAIRS 


CONCENTRATOR 

TRUNK  SW  JUNCTOR 

SW 


10 
9       C> 


TRUNK 

LINK 
FRAME 


TO  MARKER 


CONCENTRATOR    LINE   LINK 
FRAME 


Fig.  5(b)  — No.  5  crossbar  system  subscriber  lines  connected  to  remote  line 
concentrators. 


REMOTE  CONTROLLED  LINE  CONCENTRATOR 


259 


Fig.  6  —  Line  unit  construction. 


concentrator  only  in  response  to  a  pulse  in  the  associated  time  slot  or 
when  a  crosspoint  operates  (a  line  busy  pulse  is  generated  under  this 
condition  as  a  crosspoint  closure  check).  The  control  circuit  in  the 
central  office  is  designed  to  serve  10  remote  line  concentrators  connected 
to  a  single  line  link  frame.  In  this  way  the  marker  deals  with  a  concen- 
trator line  link  frame  as  it  would  with  a  regular  line  link  frame  and  the 
marker  modifications  are  minimized. 

The  traffic  loading  of  the  concentrator  is  accomplished  by  fixing  the 


260 


THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


Fig.  7(a)  —  Line  unit. 


number  of  trunks  at  10  and  equipping  or  reassigning  lines  as  needed  to 
obtain  the  trunk  loading  for  the  desired  grade  of  service.  The  six  cross- 
points,  the  passive  line  circuit  and  scanner  gates  individual  to  each  line 
are  packaged  in  one  plug-in  unit  to  facilitate  administration.  The  cross- 
points  are  placed  on  a  printed  wiring  board  together  with  a  comb  of  plug 
contacts  as  shown  in  Fig.  6.  The  entire  unit  is  then  dipped  in  rubber  and 
encapsulated  in  epoxy  resin,  as  shown  in  Fig.  7(a). 

This  portion  of  the  unit  is  extremely  reliable  and  therefore  it  may  be 
considered  as  expendable,  should  a  rare  case  of  trouble  occur.  The  passive 
line  circuit  and  scanner  gate  circuit  elements  are  mounted  on  a  smaller 
second  printed  wiring  plate  (known  as  the  "line  scanner"  plate,  see  Fig. 
7(b)  which  fits  into  a  recess  in  the  top  of  the  encapsulated  line  unit.  Cir- 


Fig.  7(b)  —  Scanner  plate  of  the  line  unit  shown  in  Fig.  7  (a). 


REMOTE    CONTROLLED    LINE    CONCENTRATOR 


261 


cuit  connection  between  printed  wiring  plates  is  through  pins  which  ap- 
pear in  the  recess  and  to  which  the  smaller  plate  is  soldered. 


6.    BASIC    CIRCUITS 


a.  Diode  Gates 


All  high  speed  signaling  is  on  a  pulse  basis.  Each  pulse  is  positive  and 
approximately  15  volts  in  amplitude.  There  is  one  basic  type  of  diode 
gate  circuit  used  in  this  equipment.  By  using  the  two  resistors,  one  con- 
denser and  one  silicon  alloy  junction  diode  in  the  gate  configuration 
shown  in  Fig.  8,  the  equivalents  of  opened  or  closed  contacts  in  relay 
circuits  are  obtained.  These  configurations  are  known  respectively  as 
enabling  and  inhibiting  gates  and  are  shown  with  their  relay  equivalents 
ill  Figs.  8(a)  and  8(b). 

In  the  enabling  gate  the  diode  is  normally  back  biased  by  more  than 
the  pulse  voltage.  Therefore  pulses  are  not  transmitted.  To  enable  or 


INPUT 


ENABLING    GATE   CIRCUIT 
CI 


OUTPUT 


(a) 


ENABLING   GATE     SYMBOL 


INPUT 


OUTPUT 


CONTROL 

EQUIVALENT    RELAY   CIRCUIT 
OUTPUT 
INPUT  f 


CONTROL 


CHli^ 


INPUT 


INHIBITING    GATE    CIRCUIT 
Cl 


OUTPUT 


INHIBITING    GATE    SYMBOL 


INPUT 


OUTPUT 


CONTROL 

EQUIVALENT   RELAY    CIRCUIT 
OUTPUT 


DhUHl 


Fig.  8  —  Gates  and  relay  equivalents. 


262  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 

open  the  gate  the  back  bias  is  reduced  to  a  small  reverse  voltage  which  is 
more  than  overcome  by  the  signal  pulse  amplitude  of  the  pulse.  The 
pulse  thus  forward  biases  the  diode  and  is  transmitted  to  the  output. 

The  inhibiting  gate  has  its  diode  normally  in  the  conducting  state  so 
that  a  pulse  is  readily  transmitted  from  input  to  output.  When  the  bias 
is  changed  the  diode  is  heavily  back  biased  so  that  the  pulse  amplitude 
is  insufficient  to  overcome  this  bias. 

The  elements  of  12  gates  are  mounted  on  a  single  printed  wiring  board 
w4th  plug-in  terminals  and  a  metal  enclosure  as  shown  in  Fig.  2(a).  All 
elements  are  mounted  in  one  side  of  the  board  so  that  the  opposite  side 
may  be  solder  dipped.  After  soldering  the  entire  unit  (except  the  plug) 
is  dipped  in  a  silicone  varnish  for  moisture  protection. 

b.  Transistor  Bistable  Circuit 

Transistors  are  inherently  well  adapted  to  switching  circuits  using  but 
two  states,  on  (saturated)  or  off.^^  In  these  circuits  with  a  current  gain 
greater  than  unity  a  negative  resistance  collector  characteristic  can  be 
obtained  which  will  enable  the  transistor  to  remain  locked  in  its  conduct- 
ing state  (high  collector  current  flowing)  until  turned  off  (no  collector 
current)  by  an  unlocking  pulse.  At  the  time  the  concentrator  develop- 
ment started  only  point  contact  transistors  were  available  in  quantity. 
Point  contact  transistors  have  inherently  high  current  gains  (>1)  but 
the  collector  current  flowing  when  in  the  normal  or  unlocked  condition 
(Ico)  was  so  great  that  at  high  ambient  temperatures  a  relay  once  op- 
erated in  the  collector  circuit  would  not  release. 

Junction  transistors  are  capable  of  a  much  greater  ratio  of  on  to  off 
current  in  the  collector  circuit.  Furthermore  their  characteristics  are 
amenable  to  theoretical  design  consideration.^^  However,  the  alpha  of  a 
simple  junction  transitor  is  less  than  unity.  To  utilize  them  as  one  would    | 
a  point  contact  transitor  in  a  negative  resistance  switching  circuit,  a 
combination  of  n-p-n  and  p-n-p  junction  transistors  may  be  employed,  i 
see  Fig.  9(b).  Two  transistors  combined  in  this  manner  constitute  a    ' 
"hooked  junction  conjugate  pairs."  This  form  of  bi-stable  circuit  was    j 
used  because  it  requires  fewer  components  and  uses  less  power  than  an 
Eccles-Jordan  bistable  circuit  arrangement.  It  has  the  disadvantage  of  a 
single  output  but  this  was  not  found  to  be  a  shortcoming  in  the  design 
of  circuits  employing  pulse  gates  of  the  type  described.  In  what  follows 
the  electrodes  of  the  transistor  will  be  considered  as  their  equivalents 
shown  in  Fig.  9(b). 

The  basic  bi-stable  circuit  employed  is  shown  in  Fig.  10.  The  set 


REMOTE    CONTROLLED    LINE    CONCENTRATOR 


263 


EMITTER 


COLLECTOR 


EMITTER 


n-p-n 


COLLECTOR 


BASE 

fa) 

POINT   CONTACT 
TRANSISTOR 

Ic 


BASE 


(b) 


CONJUGATE    PAIR 

ALLOY    JUNCTION 

TRANSISTORS 


C  _ 


0C>  1 


Fig.  9  —  Point  contact  versus  hooked  conjugate  pair. 

pulse  is  fed  into  the  emitter  (of  the  pair)  causing  the  emitter  diode  to 
conduct.  The  base  potential  is  increased  thus  increasing  the  current 
flowing  in  the  collector  circuit.  When  the  input  pulse  is  turned  off  the 
base  is  left  at  about  —2  volts  thus  maintaining  the  emitter  diode  con- 
( lucting  and  continuing  the  increased  current  flow  in  the  collector  circuit. 
The  diode  in  the  collector  circuit  prevents  the  collector  from  going 
positive  and  thereby  limits  the  current  in  the  collector  circuit.  To  reset, 
a  positive  pulse  is  fed  into  the  base  through  a  pulse  gate.  The  driving  of 
tlie  base  positive  returns  the  transistor  pair  to  the  off  condition. 

c.  Transistor  Pulse  Amplifier 

This  circuit  (Fig.  11)  is  formed  by  making  a  bi-stable  self  resetting 
circuit.  It  is  used  to  produce  a  pulse  of  fixed  duration  in  response  to  a 


TRANSISTORS 

p-n-p 


SET 


RESET 


I-5V 


-I6V 


F/F 


Fig.  10  —  Transistor  bi-stable  circuit. 


264 


THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


pulse  of  variable  width  (within  limits)  on  the  input.  Normally  the  emitter 
is  held  slightly  negative  with  respect  to  the  base.  The  potential  difference 
determines  the  sensitivity  of  the  amplifier.  When  a  positive  input  pulse 
is  received,  the  emitter  diode  conducts  causing  an  increase  in  collector 
current.  The  change  in  bias  of  the  diode  in  the  emitter  circuit  permits 
it  to  conduct  and  charge  the  condenser.  With  the  removal  of  the  input 
pulse  the  discharge  of  the  condenser  holds  the  transistor  pair  on.  The 
time  constant  of  the  circuit  determines  the  on  time.  When  the  emitter 
potential  falls  below  the  base  potential,  the  transistor  pair  is  turned  off. 

The  amplifiers  and  bi-stable  circuits  or  flip-flops,  >as  they  are  called 
more  frequently,  are  mounted  together  in  plug-in  packages.  Each  pack- 
age contains  8  basic  circuits  divided  7-1,  6-2,  or  2-6,  between  amplifiers 
and  fhp-flops.  Fig.  2(c)  shows  one  of  these  packages.  They  are  smaller 
than  the  gate  or  line  unit  packages,  having  only  28  terminals  instead  of 
42. 

The  transistors  for  the  field  trial  model  w^ere  plugged  into  small  hear- 
ing aid  sockets  mounted  on  the  printed  wiring  boards.  For  a  production 
model  it  w^ould  be  expected  that  the  transistors  w^ould  be  soldered  in. 

d.  Transistor  Ring  Counter 

By  combining  bi-stable  transistor  and  diode  pulse  gate  circuits  to- 
gether in  the  manner  shown  in  Fig.  12  a  ring  counter  may  be  made,  with 


INPUT 


p-n-p 


^w 


^vW-" 


I 

+  5V 


OUTPUT 


-16  V 


INPUT 


OUTPUT 


Fig.  11  —  Transistor  pulse  amplifier. 


REMOTE    CONTROLLED    LINE    CONCENTRATOR 


265 


COUNT 
INPUT 

lie 


STAGE   NUMBER 
3 


NOTE: 

LEADS    A-0  TO   A-4 
ARE  OUTPUT    LEADS 
OF    RESPECTIVE   STAGES 


1    I    I    \    r 

s     's     's     's      's 


Fig.  12  —  Ring  counter  schematic. 


a  bi-stable  circuit  per  stage.  The  enabling  gate  for  a  stage  is  controlled 
by  the  preceding  stage  allowing  it  to  be  set  by  an  input  advance  pulse. 
The  output  signal  from  a  stage  is  fed  back  to  the  preceding  stage  to  turn 
it  off.  An  additional  diode  is  connected  to  the  base  of  each  stage  for  re- 
setting when  returning  the  counter  to  a  fixed  reference  stage. 

A  basic  package  of  5  ring  counter  stages  is  made  up  in  the  same  frame- 
work and  with  the  same  size  plug  as  the  flip-flop  and  amplifier  packages, 
see  Fig.  2(b).  A  four  stage  ring  counter  is  also  used  and  is  the  same 
package  with  the  components  for  one  stage  omitted.  The  input  and  out- 
put terminals  of  all  stages  are  available  on  the  plug  terminals  so  that 
the  stages  may  be  connected  in  any  combination  and  form  rings  of  more 
than  5  stages.  The  reset  lead  is  connected  to  all  but  the  one  stage  which 
is  considered  the  first  or  normal  stage. 

Other  transistor  circuits  such  as  binary  counters  and  square  wave 
generators  are  used  in  small  quantity  in  the  central  office  equipment. 
They  will  not  be  described. 


266  THE    BELL    SYSTEM    TECHNICAL   JOURNAL,    MARCH    1956 


CONCENTRATOR 

LINE    BUSY 


CENTRAL  OFFICE 


TO    ALL   CROSSPOINTS 
/  SERVED    BY   TRUNK 


+  130  V 


VG 
VF 


I 


L..1.. 


/ 


TO   ALL   CROSSPOINTS 
FOR   SAME    LINE 


SELECTION 

FROM 

" CENTRAL 

OFFICE 


i-65V 


I  +  100V 


Fig.  13  —  Crosspoint  operating  circuit. 


e.  Crosspoint  Operating  Circuit 

The  crosspoint  consists  of  a  reed  relay  with  4  reed  switches  and  a  gas 
diode  (Fig.  1).  The  selection  of  a  crosspoint  is  accomplished  by  marking 
with  a  negative  potential  (  —  65  volts)  all  crosspoints  associated  with  a 
line,  and  marking  with  a  positive  potential  (  +  100  volts)  all  crosspoints 
associated  with  a  trunk  (Fig.  13).  The  line  is  marked  through  a  relay 
circuit  set  by  signals  sent  over  the  control  pair  from  the  central  office. 
The  trunk  is  marked  b}^  a  simplex  circuit  connected  through  the  break 
contacts  of  the  hold  magnet  of  the  crossbar  switch  associated  with  the 
trunk  in  the  central  office.  Only  one  crosspoint  at  a  time  is  exposed  to 
165  volts  which  is  necessary  and  sufficient  to  break  down  the  gas  diode 
to  its  conducting  state.  The  reed  relay  operates  in  series  with  the  gas 
diode.  A  contact  on  the  relay  shunts  out  the  gas  diode.  When  the  marking- 
potentials  are  removed  the  relay  remains  energized  in  a  local  30-voll 
circuit  at  the  concentrator.  The  holding  current  is  approximately  2.5  ma. 

This  circuit  is  designed  so  that  ringing  signals  in  the  presence  or  ab- 
sence of  lino  marks  will  not  falsely  fire  a  crosspoint  diode.  Furthonnoi'o, 


REMOTE    CONTROLLED    LINE    CONCENTRATOR 


267 


a  line  or  trunk  mark  alone  should  not  be  able  to  fire  a  crosspoint  diode 
on  a  busy  line  or  trunk. 

When  the  crosspoint  operates,  a  gate  which  has  been  inhibiting  pulses 
is  forward  biased  by  the  —65  volt  signal  through  the  crosspoint  relay 
winding.  The  pulse  which  initiates  the  mark  operations  at  the  concentra- 
tor then  passes  through  the  gate  to  return  a  line  busy  signal  to  the  central 
office  over  this  control  pairs  which  is  interpreted  as  a  crosspoint  closure 
check  signal. 

f.  Crosspoint  Release  Circuit 

The  hold  magnet  of  the  central  office  crossbar  switch  operates,  remov- 
ing the  +100- volt  operate  mark  signal  after  the  crosspoint  check  signal 
is  received.  A  slow  release  relay  per  trunk  is  operated  directly  by  the 
hold  magnet.  When  the  central  office  connection  in  the  No.  5  crossbar 
system  releases,  the  hold  magnet  is  released.  As  shown  in  Fig.  14,  with  the 
hold  magnet  released  and  the  slow  release  relay  still  operated,  a  — 130- 
volt  signal  is  applied  in  a  simplex  circuit  to  the  trunk  to  break  down  a 
gas  tube  provided  in  the  trunk  circuit  at  the  concentrator.  This  tube  in 


CONCENTRATOR 


CENTRAL  OFFICE 


TO  ALL  CROSSPOINTS 
SERVED  BY  SAME  TRUNK 


130V  I 


Fig.  14  —  Crosspoint  release  circuit. 


268 


THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


breaking  down  shunts  the  local  holding  circuit  of  the  crosspoint  causing 
it  to  release.  The  —  130-volt  disconnect  signal  is  applied  during  the 
release  time  of  the  slow  release  relay  which  is  long  enough  to  insure  the 
release  of  the  crosspoint  relay  at  the  concentrator. 

The  release  circuit  is  individual  to  the  trunk  and  independent  of  the 
signal  sent  over  the  control  pairs. 

g.  Pulse  Signalling  Circuits 

To  control  the  concentrator  four  distinct  pulse  signals  are  transmitted 
from  the  central  office.  Two  of  these  at  times  must  be  transmitted 
simultaneously,  bvit  these  and  the  other  two  are  transmitted  mutually 
exclusively.  In  addition,  service  request  and  line  busy  signals  are  trans- 
mitted from  the  concentrator  to  the  central  office.  The  two  way  trans- 
mission of  information  is  accomplished  on  each  pair  by  sending  signals  in 
each  direction  at  different  times  and  inhibiting  the  receipt  of  signals 
when  others  are  being  transmitted. 

To  transmit  four  signals  over  two  such  pairs,  both  positive  and  nega- 


CONTROL 
PAIR  NO.  1 


VF 


M 


LB 


D 


SR 


-16V 


VG 


CONTROL 
PAIR  NO.  2 


16  V 


M 


CONCENTRATOR 
AMPLIFIERS 


I 


CENTRAL  OFFICE 

AMPLIFIERS 

PER   CONCENTRATOR 


Fig.  15.  —  Signal  transmission  circuit. 


REMOTE    CONTROLLED    LINE    CONCENTRATOR  269 

tive  pulses  are  employed.  Diodes  are  placed  in  the  legs  of  a  center  tapped 
transformer,  as  shown  in  Fig.  15,  to  select  the  polarity  of  the  trans- 
mitted pulses.  At  the  receiving  end  the  desired  polarity  is  detected  by 
taking  the  signal  as  a  positive  pulse  from  a  properly  poled  winding  of  a 
transformer.  The  amplifier,  as  described  in  Section  6c  responds  only  to 
positive  pulses.  If  pulses  of  the  same  polarity  are  transmitted  in  the 
other  direction  over  the  same  pair,  as  for  control  pair  No.  1,  the  outputs 
of  the  receiving  amplifier  for  the  same  polarity  pulse  are  inhibited 
whenever  a  pulse  is  transmitted. 

As  shown  in  Fig.  15,  the  service  request  and  line  busy  signals  are 
transmitted  from  the  concentrator  to  the  central  office  over  one  pair  of 
conductors  as  positive  and  negative  pulses  respectivel3^  The  trans- 
mission of  these  pulses  gates  the  outputs  of  two  of  the  receiving  ampli- 
fiers at  the  concentrator  to  permit  the  receipt  of  the  polarized  signals 
from  the  central  office.  This  prevents  the  pulses  from  being  used  at  the 
sending  end.  A  similar  gating  arrangement  is  used  with  respect  to  the 
signals  when  sent  over  this  control  pair  from  the  central  office.  The  pulses 
designated  VG  or  RS  never  occur  when  a  pulse  designated  SR  or  LB 
is  sent  in  the  opposite  direction.  The  transmission  of  the  VF  pulse  over 
control  pair  No.  2  is  processed  by  the  concentrator  circuit  and  becomes 
the  SR  or  LB  pulses.  Li  section  7  the  purpose  of  these  pulses  is  described. 

The  signaling  range  objective  is  1,200  ohms  over  regular  exchange 
area  cable  including  loaded  facilities  from  sfation  to  central  office. 

h.  Power  Supply 

Alternating  current  is  supplied  to  the  concentrator  from  a  continuous 
service  bus  in  the  central  office.  The  power  supply  path  is  a  phantom 
circuit  on  the  two  control  pairs  as  shown  in  Fig.  16.  The  power  trans- 
former has  four  secondary  windings  used  for  deriving  from  bridge 
rectifiers  four  basic  dc  voltages.  These  voltages  and  their  uses  are  as 
fofiows:  —16  volts  (regulated)  for  transistor  collector  circuits  and  gate 
biases,  -|-5  volts  (regulated)  for  transistor  base  biases,  -|-30  volts  (regu- 

,  lated)  for  crosspoints  holding  circuits  and  —  65  volts  for  the  marking  and 
operating  of  the  line  crosspoints.  For  this  latter  function  a  reference  to 
the  central  office  applied  -flOO  volt  trunk  mark  is  necessary.  The  refer- 
ence ground  for  the  concentrator  is  derived  from  ground  applied  to  a 
simplex  circuit  on  the  power  supply  phantom  circuit.  Series  transistors 
and  shunt  silicon  diodes  with  fixed  reference  breakdown  voltages  are 

I  used  to  regulate  dc  voltages. 


270  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 

Total  power  consumption  of  the  concentrator  is  between  5  and  8  watts 
depending  upon  the  number  of  connections  being  held. 

7.    CONCENTRATOR    OPERATION 

a.  Line  Scanning 

The  sixty  lines  are  divided  into  12  groups  of  5  lines  each.  These  group- 
ings are  designated  VG  and  VF  respectively  corresponding  to  the 
vertical  group  and  file  designations  used  in  the  No.  5  crossbar  system. 
Each  concentrator  corresponds  to  a  horizontal  group  in  that  system. 

To  scan  the  lines  two  transistor  ring  counters,  one  of  12  stages  and 
one  of  5  stages,  are  employed  as  shown  in  Fig.  17.  These  counters  are 
driven  from  pulses  supplied  from  the  central  office  control  circuits  and 
only  one  stage  in  each  is  on  at  any  one  time.  The  steps  and  combinations 
of  these  counters  correspond  to  the  group  and  file  designation  of  a  par- 
ticular line.  Each  0.002  second  the  five  stage  counter  (VF)  takes  a 
step  and  between  the  fifth  and  sixth  pulse  the  r2-stage  counter  (VG) 
is  stepped.  Thus  the  5-stage  counter  receives  60  pulses  or  re-cycles  12 
times  in  120  milliseconds  while  the  12-stage  counter  cycles  but  once. 

Each  line  is  provided  with  a  scanner  gate.  The  collector  output  of  each 
each  stage  of  the  VG  counter  biases  this  gate  to  enable  pulses  which 
are  generated  by  the  collector  circuit  of  the  5-stage  counter  to  pass  on 


-65V 


+  30V 


+  SV 


-I6V 


115  V 
AC 


MOTOR 
GENERATOR 


TO 

COMMERCIAL 

AC 


REGULATORS 


Fig.  16  —  Power  supply  transmission  circuit. 


REMOTE    CONTROLLED   LINE   CONCENTRATOR 


271 


to  the  gate  of  the  passive  line  circuit,  Fig.  3(b).  If  the  line  is  idle  the 
pulses  are  inhibited.  If  the  receiver  is  off-hook  requesting  service  (no 
(•rosspoint  closed)  then  the  gate  is  enabled,  the  pulse  passes  to  the  service 
request  amplifier  and  back  to  the  central  office  in  the  same  time  slot 
as  the  pulse  which  stepped  the  VF  counter.  If  the  line  has  a  receiver 
off-hook  and  is  connected  to  a  trunk  the  pulse  passes  through  a  contact 
of  the  crosspoint  relay  to  the  line  busy  amplifier  and  then  to  the  central 
office  in  the  same  time  slot. 

At  the  end  of  each  complete  cycle  a  reset  pulse  is  sent  from  the  central 
office.  This  pulse  instead  of  the  VG  pulse  places  the  12-stage  counter  in 
its  first  position.  It  also  repulses  the  5  stage  VF  counter  to  its  fifth  stage 
so  that  the  next  VF  pulse  will  turn  on  its  first  stage  to  start  the  next 

j  cycle.  The  reset  pulse  insures  that,  in  event  of  a  lost  pulse  or  defect  in 
a  counter  stage,  the  concentrator  will  attempt  to  give  continuous  ser- 
\'ice  without  dependence  on  maintaining  synchronism  with  the  central 

I  office  scanner  pulse  generator.  Fig.  18(a)  shows  the  normal  sequence  of 

I  line  scanning  pulses. 

,      When  a  service  request  pulse  is  generated,  the  central  office  circuits 


t] 


04 


r 


VF  5- STAGE 
COUNTER 


03 


TO  10 

INTERMEDIATE 

GATES  EACH 

V 


02 


I 


01 


00 


TO  5  GATES  EACH 
I 


1       23456       789      10 
I       I       I       I       I       I       I       I       I       I 

I       I       I       I       I       I       I       I       I       I 


VG  12-STAGE   COUNTER 


59\ 


58 


57 


56 


55 


GATE    PER    LINE 

■  FEEDS    PASSIVE 

LINE   CIRCUITS 


/ 


VG 


RESET 
VF 


FROM 

CENTRAL 

OFFICE 


Fig.  17  —  Diode  matrix  for  scanning  lines. 


272  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 

common  to  10  concentrators  interrupt  the  further  transmission  of  the 
vertical  group  pulse  so  that  the  line  scanning  is  confined  to  the  5  lines 
in  the  vertical  group  in  which  the  call  originated.  In  this  way  the  cen- 
tral office  will  receive  a  service  request  pulse  at  least  every  0.010  sec  as 
a  check  that  the  call  has  not  been  abandoned  while  awaiting  service. 
Fig.  18(b)  shows  the  detection  of  a  call  origination  and  the  several 
short  scan  cycles  for  abandoned  call  detection. 

b.  Line  Selection 

When  the  central  office  is  ready  to  establish  a  connection  at  the  con- 
centrator a  reset  pulse  is  sent  to  return  the  counters  to  normal.  In  gen- 
eral, the  vertical  group  and  vertical  file  pulses  are  sent  simultaneously 
to  reduce  holding  time  of  the  central  office  equipment  and  to  minimize 
marker  delays  caused  by  this  operation.  For  this  reason  the  VG  and  VF 
pulses  are  each  transmitted  over  different  control  pairs  from  the  central 
office.  The  same  polarity  is  used. 

On  originating  calls  it  is  desirable  to  make  one  last  check  that  the 
call  has  not  been  abandoned,  while  on  terminating  calls  it  is  necessary 

L* 120MS >| 

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LB 
RS 


J__l I I I I I I I I I I I I I I 1 I I I I 1 I L 

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(a)  REGULAR    LINE    SCANNING 


VF 


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(b)  CALL    ORIGINATION    SERVICE    REQUEST    FROM    LINE    6/3 

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RESULTS     FROM    CONC     CONTROL  RECEIVED         'OPERATE        ""CROSSPOINT           'NORMAL  SCANNING 

CKT    AT    CENTRAL   OFFICE  ONLY     IF         CROSSPOINT           CLOSURE                           IS   RESUMED 

RECEIVING    FROM    MKR      VG ,  LINE    6/3    HAS                                     INDICATION 

VF,  HG    INFORMATION  BECOME     BUSY 


(C)  LINE    SELECTION    FOR    LINE    6/3 


Fig.  18  —  Pulse  sequences,  (a)  Regular,  (b)  Call  origination,  (c)  Line  selection. 


REMOTE    CONTROLLED    LINE    CONCENTRATOR  273 

to  determine  if  the  line  is  busy  or  idle.  These  conditions  are  determined 
in  the  same  manner  as  described  for  line  scanning  since  a  service  re- 
quest condition  would  still  prevail  on  the  line  if  the  call  was  not  aban- 
doned. If  the  line  was  busy,  a  line  busy  condition  would  be  detected. 
However  to  detect  these  conditions  a  VF  pulse  must  be  the  last  pulse 
transmitted  since  the  stepping  of  the  VF  counter  generates  the  pulse 
which  is  transmitted  through  an  enabled  line  selection  and  passive 
line  circuit  gates.  Fig.  18(c)  shows  a  typical  line  selection  where  the  num- 
ber of  VF  pulses  is  equal  to  or  less  than  the  number  of  VG  pulses.  In 
all  other  cases  there  is  no  conflict  and  the  sending  of  the  last  VF  pulse 
need  not  be  delayed.  On  terminating  calls,  the  line  busy  indication  is 
returned  to  the  central  office  within  0.002  sec  after  the  selection  is  com- 
plete. During  selections  the  central  office  circuits  are  gated  to  ignore 
any  extraneous  service  request  or  line  busy  pulses  produced  as  a  result 
of  steps  of  the  VF  counter  prior  to  its  last  step. 

c.  Crosspoint  Operation  and  Check 

Associated  with  each  concentrator  transistor  counter  stage  is  a  reed 
relay.  These  relays  are  connected  to  the  transistor  collector  circuits 
through  diodes  of  the  counter  stages  when  relay  M  operates.  The  con- 
tacts of  these  reed  relays  are  arranged  in  a  selection  circuit  as  shown 
in  Fig.  19  and  apply  the  —65  volt  mark  potential  to  the  crosspoint 
relays  of  the  selected  line. 

After  a  selection  is  made  as  described  above  a  "mark"  pulse  is  sent 
from  the  central  office.  This  pulse  is  transmitted  as  a  pulse  of  a  different 
polarity  over  the  same  control  pair  as  the  VF  pulses.  The  received 
pulse  after  amplification  actuates  a  transistor  bistable  circuit  w^hich  has 
the  M  reed  relay  permanently  connected  in  its  collector  circuit.  The 
bi-stable  circuit  holds  the  M  relay  operated  during  the  crosspoint  opera- 
tion to  maintain  one  VF  and  one  VG  relay  operated,  thereby  applying 
—  65  volts  to  mark  and  operate  one  of  the  6  crosspoint  relays  of  the 
selected  line  as  described  in  section  6e,  and  shown  on  Fig.  13. 

The  operation  and  locking  of  the  crosspoint  relay  with  the  marking 
potentials  still  applied  enables  a  pulse  gate  associated  with  the  holding 
circuit  of  the  crosspoint  relays  in  each  trunk  circuit.  The  mark  pulses 
are  sent  out  continuously.  This  does  not  affect  the  bi-stable  transistor 
circuit  once  it  has  triggered  but  the  mark  pulse  is  transmitted  through 
the  enabled  crosspoint  closure  check  gate  shown  in  Fig.  20  and  back 
to  the  central  office  as  a  line  busy  signal. 

With  the  receipt  of  the  crosspoint  closure  check  signal  the  sending 


274  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 

of  the  mark  pulses  is  stopped  and  a  reset  pulse  is  sent  to  the  concentra- 
tor to  return  the  mark  bi-stable  circuit,  counters  and  all  operated  selec- 
tor relays  to  normal.  The  concentrator  remains  in  this  condition  until 
it  is  resynchronized  with  the  regular  line  scanning  cycle. 

A  complete  functional  schematic  of  the  concentrator  integrating  the 
circuits  described  above  is  shown  in  Fig.  21.  Fig.  22(a)  and  (b)  show  an 
experimental  concentrator  built  for  field  tests. 

8.    CENTRAL   OFFICE    CIRCUITS 

The  central  office  circuits  for  controling  one  or  more  concentrators 
are  composed  of  wire  spring  relays  as  well  as  transistors,  diode  and  reed 


VG 


RS 


VF 


M 


-20V 


-20  V 
o- 


VF-5  STAGE 
COUNTER 


r 


-65V 


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6  RELAY      I        W-,  wo  w<-  ,^,  -o  p 

PACKAGE  J     '-|„p_„p_^_p-_-p-^Ui 


TO  CONTACTS  OF   4 
INTERMEDIATE   RELAYS 


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INTERMEDIATE   RELAYS 


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P^ig.  19  —  Line  selection  and  marking. 


I 


REMOTE   CONTROLLED    LINE    CONCENTRATOR 


275 


relay  packages  similar  to  those  used  in  the  concentrator.  The  reed 
relays  are  energized  by  transistor  bi-stable  circuits  in  the  same  manner 
as  described  in  Section  7c.  The  reed  relay  contacts  in  turn  operate  wire 
spring  relays  or  send  the  dc  signals  directly  to  the  regular  No.  5  crossbar 
marker  and  line  link  marker  connector  circuits. 

Fig.  23  shows  a  block  diagram  of  the  central  office  circuits.  A  small 
amount  of  circuitry  is  provided  for  each  concentrator.  It  consists  of  the 
following: 

1.  The  trunk  connecting  crossbar  switch  and  associated  slow  relays 
for  disconnect  control. 

2.  The  concentrator  control  triuik  circuits  and  associated  pulse  ampli- 
fiers. 

3.  An  originating  call  detector  to  identify  which  concentrator  among 
the  ten  served  by  the  frame  is  calling. 

4.  A  multicontact  relay  to  connect  the  circuits  individual  to  each 
concentrator  with  the  common  control  circuits  associated  with  the  line 
link  frame  and  markers. 

The  circuits  associated  with  more  than  one  concentrator  are  blocked 
out  in  the  lower  portion  of  Fig.  23.  Much  of  this  circuitry  is  similar  to 
the  relay  circuits  now  provided  on  regular  line  link  frames  in  the  No.  5 
crossbar  system.^  Only  those  portions  of  these  blocks  which  employ  the 
new  techniques  will  be  covered  in  more  detail.  These  portions  consist 
of  the  following: 

1.  The  scanner  pulse  generator. 

2.  The  originating  line  number  register. 


T 


TO  ALL   TRUNK   LINES 
+  30V 


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Fig.  20  —  Crosspoint  closure  check. 


aoidzio  nvbiNBo  oi 
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276 


Fig.  22(a)  —  Complete  line  concentrator  unit. 

r 5 -STAGE   COUNTER 

12 -STAGE  COUNTER 


-fO  TRUNK   CiftCUITS 

AMPLIFIERS 
RECTIFIERS 


Fig.  22(b)  —  Identification  of  units  within  the  line  concentrator. 

277 


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REMOTE   CONTROLLED    LINE   CONCENTRATOR 


279 


3.  The  line  selection  circuit. 

4.  The  trunk  identifier  and  selection  relay  circuits. 

(For  an  understanding  of  how  these  frame  circuits  work  through  the  line 
(link  marker  connector  and  markers  in  the  No.  5  system,  the  reader 
should  consult  the  references.) 

The  common  central  office  circuits  will  be  described  first. 


a.  Scanner  Pulse  Generator 

The  scanner  pulse  generator,  shown  in  Fig.  24,  produces  continuously 
the  combination  of  VG,  VF  and  RS  or  reset  pulses,  described  in  connec- 
tion with  Fig.  18(a),  required  to  drive  the  scanners  for  a  number  of 
concentrators.  The  primary  pulse  source  is  a  1,000-cycle  transistor 
oscillator.  This  oscillator  drives  a  transistor  bi-stable  circuit  arranged 
as  a  binary  counter  such  that  on  each  cycle  of  the  oscillator  output  it 
alternately  assumes  one  of  its  states.  Pulses  produced  by  one  state  drive 
a  5-stage  counter.  Pulses  produced  by  the  other  state  through  gates 
drive  a  12-stage  counter. 

The  pulses  which  drive  the  5-stage  counter  are  the  same  pulses  which 
are  used  for  the  VF  pulses  to  drive  scanners.  Each  time  the  first  stage 
of  the  5-stage  counter  is  on,  a  gate  is  opened  to  allow  a  pulse  to  drive 
the  12-stage  counter.  The  pulses  which  drive  the  12-stage  counter  are 
also  the  pulses  used  as  the  VG  pulses  for  driving  the  scanners.  They 
are  out  of  phase  with  the  VF  pulses. 

When  the  last  stage  of  the  12-stage  counter  is  on,  the  gate  which 


r  VFC 


Fig.  24  —  Scanner  pulse  generator. 


280 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    195(5 


transmits  pulses  to  the  12-stage  counter  is  closed  and  another  gate  is 
opened  which  produces  the  reset  pulse.  The  reset  pulse  is  thereby  trans- 
mitted to  the  scanners  in  place  of  the  first  vertical  group  pulse.  At  the 
same  time  the  5  and  12-stage  counters  in  the  scanner  pulse  generator 
are  reset  to  enable  the  starting  of  a  new  cycle. 

In  the  central  office  control  circuits,  out  of  phase  pulses  on  lead  TP 
similar  to  those  which  drive  the  VG  counters  at  the  concentrator  are 
used  for  various  gating  operations. 

b.  The  Originating  Call  Detection  and  Line  Number  Registration 

The  originating  call  detector  (Fig.  25)  and  the  originating  line  num- 
ber register  (Fig.  26)  together  receive  the  information  from  the  line 
concentrator  used  to  identify  the  number  of  the  line  making  a  service  i 
request.  The  receipt  of  the  service  request  pulse  from  a  concentrator  i 
in  a  particular  time  slot  will  set  a  transistor  bi-stable  circuit  HGT  of  { 
Fig.  25  associated  with  that  concentrator  if  no  other  originating  call  is 
being  served  by  the  frame  circuits  at*  this  time. 

The  originating  line  number  register  consists  of  a  5  and  12-stage 
counter.  These  counters  are  normally  driven  through  gates  in  syn- 
chronism with  the  scanning  counters  at  concentrators  with  pulses  sup- 
plied from  the  scanner  pulse  generator.  When  a  service  request  pulse 
is  received  from  any  of  the  concentrators  served  by  a  line  link  frame,  a 
pulse  is  sent  to  the  originating  line  number  register  which  operates  a 
bi-stable  circuit  over  a  lead  RH  in  Fig.  26.  This  bi-stable  circuit  then 
closes  the  gates  through  which  the  5-  and  12-stage  counters  are  being 
driven,  and  also  closes  a  gate  which  prevents  them  from  being  reset. 


TO   TRAFFIC  I 
RECORDER   I 


TO  ORIGINATING 
CALL    REGISTER 


I  TO   CONCENTRATOR 
I     CONTROL  TRUNK 


Fig.  25  —  Originating  call  detector. 


EEMOTE    CONTROLLED    LINE    CONCENTRATOR 


281 


In  this  way,  the  number  of  the  line  which  originated  a  service  request  is 
locked  into  these  counters  until  the  bi-stable  circuit  is  restored  to  nor- 
mal. 

The  HGT  bi-stable  circuit  of  Fig.  25  indicates  which  particular  con- 
centrator has  originated  a  service  request.  A  relay  in  the  collector  cir- 
cuit has  contacts  which  pass  this  information  on  to  the  other  central 
office  control  circuits  to  indicate  the  number  of  the  concentrator  on  the 
frame  which  is  requesting  service.  This  is  the  same  as  a  horizontal  group 
on  a  regular  line  link  frame  and  hence  the  horizontal  group  designation 
is  used  to  identify  a  concentrator. 

With  the  operation  of  this  relay,  relays  associated  with  the  counters 
of  the  originating  line  number  register  are  operated.  These  relays  indicate 
to  the  other  central  office  circuits  the  vertical  file  and  vertical  group 
identification  of  the  calling  line.  Contacts  on  the  vertical  group  relays 
are  used  to  set  a  bi-stable  circuit  associated  with  lead  RL  of  Fig.  25  each 
time  the  scanner  pulse  generator  generates  a  pulse  corresponding  to  the 
vertical  file  of  the  calling  line  number  registered. 

The  operation  of  the  HGT  bi-stable  circuit  inhibits  in  the  concentra- 
tor control  trunk  circuit  (Fig.  27)  the  transmission  of  further  VG  and 

SRS 


FROM 
CONCENTRATOR 

CONTROL 
TRUNK  CIRCUIT 


RB 
RH 


RH 


FROM 
SCANNER 

PULSE 
GENERATOR 


VF 


VFO-4 
VG 


RS 


1 


*"        5-STAGE  COUNTER        ^ 


12-STAGE   COUNTER 


^        ^ 


Fig.  26  —  Originating  line  number  register. 


282  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 

reset  pulses  to  the  concentrator  so  that,  as  described  in  Section  7a, 
only  the  VF  counter  continues  to  step  once  each  0.010  sec.  So  long  as 
the  line  continues  to  request  service  this  service  request  pulse  is  gated 
to  reset  the  RL  bi-stable  circuit  within  the  same  time  slot  that  it  was 
set.  If,  however,  a  request  for  service  is  abandoned  the  RL  bi-stable  cir- 
cuit of  Fig.  26  will  remain  on  and  permit  a  TP  pulse  from  the  scanner 
pulse  generator  to  reset  the  HGT  bi-stable  circuit  which  initiated  the 
service  request  action. 

Whenever  the  RH  bi-stable  circuit  of  Fig.  26  is  energized  it  closes  a 
gate  over  lead  SRS  for  each  concentrator  to  prevent  any  further  service 
request  pulses  from  being  recognized  until  the  originating  call  which 
has  been  registered  is  served.  The  resetting  of  the  RH  bi-stable  circuit 
occurs  once  the  call  has  been  served.  When  more  than  one  line  concen- 
trator is  being  served  it  is  possible  that  the  HGT  bi-stable  circuit  of 
more  than  one  concentrator  will  be  set  simultaneously  as  a  result  of 
coincidence  in  service  requests  from  correspondingly  numbered  lines  in 
these  concentrators.  The  decision  as  to  which  concentrator  is  to  be 
served  is  left  to  the  marker,  as  it  would  normally  decide  which  horizontal 
group  to  serve. 

c.  Line  Selection 

On  all  calls,  originating  and  terminating,  the  marker  transmits  to  the 
frame  circuits  the  complete  identity  of  the  line  which  it  will  serve.  In 
the  case  of  originating  calls  it  has  received  this  information  in  the  manner 
described  in  Section  8b.  In  either  case,  it  operates  wire  spring  relays 
VGO-U  and  VFO-4,  which  enable  gates  so  that  the  information  may  be 
stored  in  the  5-  and  12-stage  counters  of  the  line  selection  circuit  shown  " 
in  Fig.  28. 

The  process  of  reading  into  the  line  selection  counters  starts  when 
selection  information  has  been  received  by  the  actuation  of  the  HGS 
bi-stable  circuit  in  the  concentrator  control  trunk  circuit  of  Fig.  27. 
This  action  stops  the  regular  transmission  of  scanner  pulses  if  they 
have  not  been  stopped  as  a  result  of  a  call  origination.  At  the  same  time 
it  enables  gates  for  transmission  of  information  from  the  line  selection 
circuit.  Fig.  28. 

The  ST  bi-stable  circuit  of  the  line  selection  circuit  is  also  enabled 
to  start  the  process  of  setting  the  line  selection  counters.  The  next  TP 
pulse  sets  the  Rl  bi-stable  circuit.  This  bi-stable  circuit  enables  a  gate 
which  permits  the  next  TP  pulse  to  set  the  counters  and  transmit  a  re- 
set pulse  to  the  concentrator  through  pulse  amplifier  RIA.  At  the  same 
time  bi-stable  circuit  ST  is  reset  to  prevent  the  further  read-in  cr  reset 


\ 


REMOTE   CONTROLLED   LINE    CONCENTRATOR 


283 


pulses  and  to  permit  pulses  through  amplifier  OPA  to  start  the  out- 
pulsing  of  line  selections.  These  pulses  pass  to  the  VGP  and  VFP  leads 
as  long  as  the  VG  and  VF  line  selection  counters  have  not  reached 
their  first  and  last  stages  respectively.  The  output  pulses  to  the  con- 
centrator are  also  fed  into  the  drive  leads  of  these  counters  so  that,  as 
the  counters  in  the  concentrator  are  stepped  up,  the  counters  in  the 
central  office  line  selection  circuit  are  stepped  down.  When  the  first 
stage  of  the  VF  counter  goes  on,  the  VF  pulses  are  no  longer  transmitted 
until  the  first  stage  of  the  VG  counter  goes  on.  This  insures  that  a  VF 
pulse  is  the  last  to  be  transmitted.  Also  this  pulse  is  not  transmitted 
until  the  other  frame  circuits  have  successfully  completed  selections  of 
an  idle  concentrator  trunk.  Then  bi-stable  circuit  VFLD  is  energized, 


TO   ORIGINATING 

CALL   DETECTOR 

I 


VF- 

FROM 
VG   U    SCANNER 
PULSE 
GENERATOR 


FROM    LINE 
[-SELECTION 
CIRCUIT 


Fig.  27  —  Concentrator  control  trunk  circuit. 


284 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


producing,  during  its  transition,  the  last  VF  pulse  for  transmission  to 
the  concentrator. 


d.  Trunk  Selection  and.  Identification 

The  process  of  selecting  an  idle  concentrator  trunk  to  which  the  line 
has  access  utilizes  familar  relay  circuit  techniques.^^  This  circuit,  in 
Fig.  29,  will  not  be  described  in  detail.  One  trunk  selection  relay,  TS,  is 
operated  indicating  the  preferred  idle  trunk  serving  a  line  in  the  particu- 
lar vertical  group  being  selected  as  indicated  by  the  VG  relay  which 
has  been  operated  by  the  marker. 

The  TS4  and  TS5  relays  select  trunks  8  and  9  which  are  available  to 
each  line  while  the  4  trunks  available  to  only  half  of  the  lines  are  selected 
by  relays  TS0-TS3.  The  busy  or  idle  condition  of  each  trunk  is  indicated 
by  a  contact  on  the  hold  magnet  associated  with  each  trunk  through 


TRUNK 
SELECTION 
COMPLETE  T 


VFLD 


_l 
O 

cr 

I- 
z 
Oh 

^5 
tr  u 
O  cr 


Z  : 

LU  : 

^! 

o 
o 

o 

t- 


VFP 


VGP_ 
RS 


VFLI 


VFL  2 


0-1 


5-STAGE   COUNTER 


VF4X  -2V 


VGO 


y^ 


12-STAGE   COUNTER 


0-0 


ST 


OPA 


R1A 


ST 


FROM   SCANNER 
PULSE   GENERATOR  I 


TP 


Fig.  28  —  Line  selection  circuit. 


REMOTE    CONTROLLED    LINE    CONCENTRATOR 


285 


relay  HG  which  operates  on  all  originating  and  terminating  calls  to  the 
particular  concentrator  served  by  these  trunks.  The  end  chain  relay 
TC  of  the  lockout  trunk  selection  circuit^^  connects  battery  from  the 
SR  relay  windings  of  idle  trunks  to  the  windings  of  the  TS  relays  to 
permit  one  of  the  latter  relays  to  operate  and  to  steer  circuits,  not  shown 
on  Fig.  29,  to  the  hold  magnet  of  the  trunk  and  to  the  tip-and-ring  con- 
ductors of  the  trunk  to  apply  the  selection  voltages  shown  on  Figs.  13 
and  14. 

The  path  for  operating  the  hold  magnet  originates  in  the  marker. 
The  path  looks  like  that  which  the  marker  uses  on  the  line  hold  mag- 
net when  setting  up  a  call  on  a  regular  line  link  frame.  For  this  reason 
and  other  similar  reasons  this  concentrator  line  link  frame  concept  has 
been  nicknamed  the  "fool-the-marker"  scheme. 

Should  a  hold  magnet  release  while  a  new  call  is  being  served  the 
ground  from  the  TC  relaj^  normal  or  the  TS  relay  winding  holds  relay 


CONCENTRATOR    TRUNK 
SWITCH   CROSSPOINTS 


SR  [ 


LINE    LINK 
NUMBER    FROM 
MARKER  I 


-48  V 


Fig.  29  —  Trunk  selection  and  identification. 


286  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    1956 


SR  operated  through  its  own  contact  until  the  new  call  has  been  set  up. 
This  prevents  interference  of  disconnect  pulses  applied  to  the  trunk 
when  a  selection  is  being  made  and  insures  that  a  disconnect  pulse  is 
transmitted  before  the  trunk  is  reused. 

A  characteristic  of  the  No.  5  crossbar  system  is  that  the  originating 
connection  to  a  call  register  including  the  line  hold  magnet  is  released 
and  a  new  connection,  known  as  the  "call  back  connection",  is  estab- 
lished to  connect  the  line  to  a  trunk  circuit  after  dialing  is  completed. 

With  concentrator  operation  the  concentrator  trunk  switch  connection 
is  released  but  the  disconnect  signal  is  not  sent  to  the  concentrator  as 
a  result  of  holding  the  SR  relay  as  described  above.  However,  the  marker 
does  not  know  to  which  trunk  the  call  back  connection  is  to  be  estab- 
lished. For  this  reason  the  frame  circuits  include  an  identification  proc- 
ess for  determining  the  number  of  the  concentrator  trunk  to  be  used 
on  call  back  prior  to  the  release  of  the  originating  register  connection,  i 

Identification  is  accomplished  by  the  marker  transmitting  to  the 
frame  circuits  the  number  of  the  link  being  used  on  the  call.  This  in- 
formation is  already  available  in  the  No.  5  system.  The  link  being  used 
is  marked  with  —48  volts  by  a  relay  selecting  tree^"  to  operate  the  TS 
relay  associated  with  the  trunk  to  which  the  call  back  connection  is  to 
be  established.  Relay  CB  (Fig.  29)  is  operated  on  this  type  of  call  in- 
stead of  relay  HG.  The  circuits  for  reoperating  the  proper  hold  magnet 
are  already  available  on  the  TS  relay  which  was  operated,  thereby  rc- 
selecting  the  trunk  to  which  the  customer  is  connected.  The  concen- 
trator connection  is  not  released  when  the  hold  magnet  releases  and 
again  the  marker  operates  as  it  would  on  a  regular  line  link  frame  call. 

9.    FIELD   TRIALS 

Three  sets  of  the  experimental  equipment  described  here  have  been 
constructed  and  placed  in  service  in  various  locations.  The  equipment 
for  these  trials  is  the  forerunner  of  a  design  for  production  which  will 
incorporate  device,  circuit  and  equipment  design  changes  based  on  the 
trial  experiences.  Fig.  30  shows  the  cabinet  mounted  central  office  trial 
equipment  with  the  designation  of  appropriate  parts. 

For  the  field  trials  described,  the  line  links  on  a  particular  horizontal 
level  of  existing  line  link  frames  were  extended  to  a  separate  cross-bar 
switch  provided  for  this  purpose  in  the  trial  equipment.  The  regular  line 
link  connector  circuits  were  modified  to  work  with  the  trial  control 
circuits  whenever  a  call  was  originated  or  terminated  on  this  level.  N(i 
lines  were  terminated  in  the  regular  primary  line  switches  for  this  level. 


REMOTE    CONTROLLED    LINE    CONCENTRATOR 


287 


10.    MISCELLANEOUS   FEATURES   OF   TRIAL   EQUIPMENT 

There  are  a  number  of  auxiliary  circuits  provided  with  the  trial  equip- 
ment to  aid  in  the  solutions  of  problems  brought  about  by  the  concepts 
of  concentrator  service.  One  of  the  purposes  of  the  trials  was  to  deter- 
mine the  way  in  which  the  various  traffic,  plant  and  commercial  ad- 


CONCENTRATOR 
TRUNK  SWITCH 


SERVICE   OBSERVING 

TEST  CONTROL-] 

SIMULATOR 

TRUNK    DISCONNECT 
RELAYS 


CONCENTRATOR 
CONNECTOR  RELAYS 


FRAME   RELAY 
CIRCUITS 


SERVICE   DENIAL 


FRAME    ELECTRONIC 
CIRCUITS 


POWER   SUPPLY 


LINE   CONDITION 
TESTER 


Fig.  30  —  Trial  central  office  equipment. 


288  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 

ministrative  functions  could  be  economically  performed  when  concen- 
trators become  common  telephone  plant  facilities.  The  more  important 
of  these  miscellaneous  features  are  discussed  under  the  following  head- 
ings : 

a.  Traffic  Recording 

J 
To  measure  the  amount  and  characteristics  of  the  traffic  handled  by 

the  concentrator  a  magnetic  tape  recorder,  Fig.  31,  was  provided  for 

each  trial.  The  number  of  the  lines  and  trunks  in  use  each  15  seconds 

during  programmed  periods  of  each  day  were  recorded  in  coded  form 

with  polarized  pulses  on  the  3-track  magnetic  tape  moving  at  a  speed  of 

1}/2"  per  second.  Combinations  of  these  pulses  designate  trunks  busy  on 

intra-concentrator  connections  and  reverting  calls. 

The  line  busy  indications  were  derived  directly  from  the  line  busy 
information  received  during  regular  scanning  at  the  concentrator.  Dur- 
ing one  cycle  in  each  15  seconds  new  service  requests  were  delayed  to 
insure  that  a  complete  scan  cycle  would  be  recorded.  Terminating  calls 
were  not  delayed  since  marker  holding  time  is  involved.  Trunk  condi- 
tions are  derived  for  a  trunk  scanner  provided  in  the  recorder. 

In  addition  to  recording  the  line  and  trunk  usage,  recordings  were 
made  on  the  tape  for  each  service  request  detected  during  a  programmed 
period  to  measure  the  speed  with  which  each  call  received  dial  tone 
and  the  manner  in  which  the  call  was  served.  In  this  type  of  operation 
the  length  of  the  recording  for  each  request  made  at  a  tape  speed  of 
only  \i!'  per  second  is  a  measure  of  service  delay  time. 

As  may  be  observed  from  Fig.  31  the  traffic  recorder  equipment  was 
built  with  vacuum  tubes  and  hence  required  a  rather  large  power  supply. 
It  is  expected  that  a  transistorized  version  of  this  traffic  recorder  serv- 
ing all  concentrators  in  a  central  office  will  be  included  in  the  standard 
model  of  the  line  concentrator  equipment.  With  this  equipment,  traffic 
engineers  will  know  more  precisely  the  degree  to  which  each  concentra-- 
tor  may  be  loaded  and  hence  insure  maximum  utilization  of  the  concen- 
trator equipment. 

b.  Line  Condition  Tester 

It  has  been  a  practice  in  more  modern  central  office  equipment  to 
include  automatic  line  testing  equipment.^^  An  attempt  has  been  made 
to  include  similar  features  with  the  concentrator  trial  equipment.  The 
line  condition  tester  (see  Fig.  30)  provides  a  means  for  automatically 
connecting  a  test  circuit  to  each  line  in  turn  once  a  test  cycle  has  been 


I 


REMOTE    CONTROLLED    LINE    CONCENTRATOR 


289 


!    ,        P  ]  'f 


^  u 


POWER  SUPPLIES 

AND 

PROGRAMMER 


I 


Fig.  31  —  Traffic  recorder. 


290  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 

manually  initiated.  This  test  is  set  up  on  the  basis  of  the  known  concen- 
trator passive  line  circuit  capabilities.  Should  a  line  fail  to  pass  this 
test,  the  test  circuit  stops  its  progress  and  brings  in  an  alarm  to  summon 
central  office  maintenance  personnel.  The  facilities  of  the  line  tester  are 
also  used  to  establish,  under  manual  control,  calls  to  individual  lines  as 
required  to  carry  out  routine  tests. 

c.  Simulator 

As  the  central  office  sends  out  scanner  control  pulses  either  no  signal, 
a  line  busy  or  service  request  pulse  is  returned  to  the  central  office  in 
each  time  slot.  The  simulator  test  equipment,  shown  in  Fig.  30,  was 
designed  to  place  pulses  in  a  specific  time  slot  to  simulate  a  line  under 
test  at  the  concentrator. 

In  addition  to  transmitting  the  equivalent  of  concentrator  output 
pulses  the  simulator  can  receive  the  regular  line  selection  pulses  trans- 
mitted to  the  concentrator  for  purposes  of  checking  central  office  opera- 
tions. It  is  possible  by  combined  use  of  the  line  tester  and  simulator  to 
observe  the  operation  of  the  concentrator  and  to  determine  the  probable 
cause  when  a  fault  occurs. 

d.  Service  Observing 

The  removal  of  the  line  terminals  from  the  central  office  poses  a  num- 
ber of  problems  in  conjunction  with  the  administration  of  central  office 
equipment.  One  of  these  is  service  observing. 

To  maintain  a  check  on  the  quality  of  service  being  rendered  by  the 
telephone  system,  service  observing  taps  are  made  periodically  on  tele- 
phone lines.  This  is  normally  done  by  placing  special  connector  shoes 
on  line  terminations  in  the  central  office. 

To  place  such  shoes  at  the  remote  concentrator  point  would  lead  to 
administrative  difficulties  and  added  expense.  Therefore,  a  method  was 
devised  to  permit  service  observing  equipment  to  be  connected  to  con- 
centrator trunks  on  calls  from  specific  lines  which  were  to  be  observed. 
This  mcithod  consisted  of  manual  switches  on  which  were  set  the  number 
of  the  line  to  be  observed  in  terms  of  vertical  group  and  vertical  file. 
Whenever  this  line  originated  a  call  and  the  call  could  be  placed  over  the 
first  preferred  trunk,  automatic  connection  was  made  to  the  service  ob- 
serving desk  in  the  same  manner  as  would  occur  for  a  line  terminated 
directly  in  the  central  office. 

In  addition,  facilities  were  provided  for  trying  a  new  service  observ- 
ing technique  where  calls  originating  over  a  particular  concentrator 


REMOTE    CONTROLLED   LINE   CONCENTRATOR  291 

trunk  would  be  observed  without  knowledge  of  the  originating  line  num- 
ber. For  this  purpose  a  regular  line  observing  shoe  was  connected  to 
one  of  the  ten  concentrator  trunk  switch  verticals  in  the  trial  equipment 
and  from  here  connected  to  the  service  observing  desk  in  the  usual 
manner. 

The  basic  service  observing  requirements  in  connection  with  line 
concentrator  operation  have  not  as  yet  been  fully  determined.  How- 
ever, it  appears  at  this  time  that  the  trunk  observing  arrangement  may 
be  preferable. 

e.  Service  Denial 

In  most  systems  denial  of  originating  service  for  non-payment  of 
telephone  service  charges,  for  trouble  interception  and  for  permanent 
signals  caused  by  cable  failures  or  prolonged  receiver-off-hook  conditions 
may  be  treated  by  the  plant  forces  at  the  line  terminals  or  by  blocking 
the  line  relay.  To  avoid  concentrator  visits  and  to  enable  the  prompt 
clearing  of  trouble  conditions  which  tie  up  concentrator  trunks,  a  ser- 
vice denial  feature  has  been  included  in  the  design  of  the  central  office 
circuits. 

This  feature  consists  of  a  patch-panel  with  special  gate  cords  which 
respond  to  particular  time  slots  and  inhibit  service  request  signals  pro- 
duced by  a  concentrator  during  this  period.  In  this  way  service  requests 
can  be  ignored  and  prevent  originating  call  service  on  particular  lines 
until  a  trouble  locating  or  other  administrative  procedure  has  been 
invoked. 

f.  Display  Circuit 

A  special  electronic  switch  was  developed  for  an  oscilloscope.  This 
arrangement  permited  the  positioning  of  line  busy  and  service  request 
pulses  in  fixed  positions  representing  each  of  the  60  lines  served.  Line 
busy  pulses  were  shown  as  positive  and  service  request  pulses  as  negative. 
This  plug  connected  portable  aid,  see  Fig.  32,  was  useful  in  tracing  calls 
and  identifying  lines  to  which  service  may  be  denied,  due  to  the  existence 
of  permanent  signals. 

Other  circuits  and  features,  too  detailed  to  be  covered  in  this  paper, 
have  been  designed  and  used  in  the  field  trials  of  remote  line  concen- 
trators. Much  has  been  learned  from  the  construction  and  use  of  this 
equipment  which  will  aid  in  making  the  production  design  smaller, 
lighter,  economical,  serviceable  and  reliable. 

Results  from  the  field  trials  have  encouraged  the  prompt  undertaking 


292  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


Fig.  32  —  Pulse  display  oscilloscope. 


REMOTE    CONTROLLED    LINE    CONCENTRATOR  293 

of  development  of  a  remote  line  concentrator  for  quantity  production. 
The  cost  of  remote  line  concentrator  equipment  will  determine  the  ul- 
timate demand.  In  the  meantime,  an  effort  is  being  made  to  take  advan- 
tage of  the  field  trial  experiences  to  reduce  costs  commensurate  with 
insuring  reliable  service. 

The  author  wishes  to  express  his  appreciation  to  his  many  colleagues 
at  Bell  Telephone  Laboratories  whose  patience  and  hard  work  have 
been  responsible  for  this  new  adventure  in  exploratory  switching  de- 
velopment. An  article  on  line  concentrators  would  not  be  complete 
without  mention  of  C.  E.  Brooks  who  has  encouraged  this  development 
and  under  whose  direction  the  engineering  studies  were  made. 

BIBLIOGRAPHY 

1.  E.  C.  Molina,  The  Theory  of  Probabilities  Applied  to  Telephone  Trunking 

Problems,  B.S.T.J.,  1,  pp.  69-81,  Nov.,  1922. 

2.  Strowger  Step-bv-Step  System,  Chapter  3,  Vol.  3,  Telephone  Theory  and 

Practice  by  K.B.  Miller.  McGraw-Hill  1933. 

3.  F.  A.  Korn  and  J.  G.  Ferguson,  Number  5  Crossbar  Dial  Telephone  Switching 

System,  Elec.  Engg.,  69,  pp.  679-684,  Aug.,  1950. 

4.  U.S.  Patent  1,125,965. 

5.  O.  Myers,  Common  Control  Telephone  Switching  Systems,  B.S.T.J.,  31,  pp. 

1086-1120,  Nov.,  1952. 

6.  L.  J.  Stacy,  Calling  Subscribers  to  the  Telephone,  Bell  Labs.  Record,  8,  pp. 

113-119,  Nov.,  1929. 

7.  J.  Meszar,  Fundamentals  of  the  Automatic  Telephone  Message  Accounting 

System,  A.  I.  E.  E.  Trans.,  69,  pp.  255-268,  (Part  1),  1950. 

8.  O.  M.  Hovgaard  and  G.  E.  Perreault,  Development  of  Reed  Switches  and 

Relays,  B.S.T.J.,  34,  pp.  309-332,  Mar.,  1955. 

9.  W.  A.  Malthaner  and  H.  E.  Vaughan,  Experimental  Electronically  Controlled 

Automatic  Switching  System,  B.  S.T.J.,  31,  pp.  443-468,  May,  1952. 

10.  S.  T.  Brewer  and  G.  Hecht,  A  Telephone  Switching  Network  and  its  Electronic 

Controls,  B.S.T.J.,  34,  pp.  361-402,  Mar.,  1955. 

11.  L.  W.  Hussey,  Semiconductor  Diode  Gates,  B.S.T.J.,  32,  pp.  1137-54,  Sept., 

1953. 

12.  U.  S.  Patent  1,528,982. 

13.  J.  J.  EbersandS.  L.  Miller,  Design  of  Alloyed  Junction  Germanium  Transis- 

tor for  High-Speed  Switching,  B. S.T.J. ,  34,  pp.  761-781,  July,  1955. 

14.  W.  B.  Graupner,  Trunking  Plan  for  No.  5  Crossbar  System,  Bell  Labs.  Record, 

27,  pp.  360  365,  Oct.,  1949. 

15.  G.  L.  Pearson  and  B.  Sawyer,  Silicon  p-n  Junction  Alloy  Diodes,  I.R.E.  Proc, 

42,  pp.  1348-1351,  Nov."  1952. 

16.  A.  E.  Anderson,  Transistors  in  Switching  Circuits,  B.S.T.J.,  31,  pp.  1207- 

1249,  Nov.,  1952. 

17.  J.  J.  Ebers  and  J.  L.  Moll,  Large-Signal  Behavior  of  Junction  Transistors, 

I.  R.  E.  Proc,  42,  pp.  1761-1784,  Dec,  1954. 

18.  J.  J.  Ebers,  Four-Terminal  p-n-p-n  Transistors,  I.  R.  E.  Proc,  42,  pp.  1361- 

1364,  Nov.,  1952. 

19.  A.  E.  Joel,  Relay  Preference  Lockout  Circuits  in  Telephone  Switching,  Trans. 

A.  L  E.  E.,  67,  pp.  720-725,  1948. 

20.  S.  H.  Washburn,  Relay  "Trees"  and  Symmetric  Circuits,  Trans.  A.  I.  E.  E., 

68,  pp.  571-597,  1949. 

21.  J.  W.  Dehn  and  R.  W.  Burns,  Automatic  Line  Insulation  Testing  Equipment 

for  Local  Crossbar  Systems,  B.S.T.J.,  32,  pp.  627-646,  1953. 


Transistor  Circuits  for  Analog  and 
Digital  Systems* 

By  FRANKLIN  H.  BLECHER 

(Manuscript  received  November  17,  1955) 

This  paper  describes  the  application  of  junction  transistors  to  precision 
circuits  for  use  in  analog  computers  and  the  input  and  output  circuits  of 
digital  systems.  The  three  basic  circuits  are  a  summing  amplifier,  an  inte- 
grator, and  a  voltage  comparator.  The  transistor  circuits  are  combined  into 
a  voltage  encoder  for  translating  analog  voltages  into  equivalent  time  inter- 
vals. 

1.0.   INTRODUCTION 

Transistors,  because  of  their  reliability,  small  power  consumption, 
and  small  size  find  a  natural  field  of  application  in  electronic  computers 
and  data  transmission  systems.  These  advantages  have  already  been 
realized  by  using  point  contact  transistors  in  high  speed  digital  com- 
puters. This  paper  describes  the  application  of  junction  transistors  to 
precision  circuits  which  are  used  in  dc  analog  computers  and  in  the 
input  and  output  circuits  of  digital  systems.  The  three  basic  circuits 
which  are  used  in  these  applications  are  a  summing  amplifier,  an  inte- 
grator, and  a  voltage  comparator.  A  general  procedure  for  designing 
these  transistor  circuits  is  given  with  particular  emphasis  placed  on  new 
design  methods  that  are  necessitated  by  the  properties  of  junction 
transistors.  The  design  principles  are  illustrated  by  specific  circuits. 
The  fundamental  considerations  in  the  design  of  transistor  operational 
amplifiers  are  discussed  in  Section  2.0.  In  Section  3.0  an  illustrative 
summing  amplifier  is  described,  which  has  a  dc  accuracy  of  better  than 
one  part  in  5,000  throughout  an  operating  temperature  range  of  0  to 
50°C.  The  feedback  in  this  amplifier  is  maintained  over  a  broad  enough 
frequency  band  so  that  full  accuracy  is  attained  in  about  100  micro- 
seconds. 

The  design  of  a  specific  transistor  integrator  is  presented  in  Section 

*  Submitted  in  partial  fulfillment  of  the  requirements  for  the  degree  of  Doctor 
of  Electrical  Engineering  at  the  Polytechnic  Institute  of  Brooklyn. 

295 


296  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    1956 


I 


4.0.  The  integrator  can  be  used  to  generate  a  voltage  ramp  which  is 
linear  to  within  one  part  in  8,000.  By  means  of  an  automatic  zero  set 
(AZS)  circuit  which  uses  a  magnetic  detector,  the  slope  of  the  voltage 
ramp  is  maintained  constant  to  within  one  part  in  8,000  throughout  a 
temperature  range  of  20°C  to  40°C. 

The  voltage  comparator,  described  in  Section  5.0,  is  an  electrical  de- 
vice which  indicates  the  instant  of  time  an  input  voltage  waveform 
passes  through  a  predetermined  reference  level.  By  taking  advantage 
of  the  properties  of  semiconductor  devices,  the  comparator  can  be  de- 
signed to  have  an  accuracy  of  ±5  millivolts  throughout  a  temperature 
range  of  20°C  to  40°C. 

In  Section  6.0,  the  system  application  of  the  transistor  circuits  is 
demonstrated  by  assembling  the  summing  amplifier;  the  integrator,  and 
the  voltage  comparator  into  a  voltage  encoder.  The  encoder  can  be  used  J 
to  translate  an  analog  input  voltage  into  an  equivalent  time  interval 
with  an  accuracy  of  one  part  in  4,000.  This  accuracy  is  realized  through- 
out a  temperature  range  of  20°C  to  40°C  for  the  particular  circuits 
described. 

2.0.   FUNDAMENTAL    CONSIDERATIONS    IN    THE    DESIGN    OF    OPERATIONAL 
AMPLIFIERS 

The  basic  active  circuit  used  in  dc  analog  computers  is  a  direct  coupled 
negative  feedback  amplifier.  With  appropriate  input  and  feedback  net- 
works, the  amplifier  can  be  used  for  multiplication  by  a  constant  coef- 
ficient, addition,  integration,  or  differentiation  as  shown  in  Figure  1 
The  accuracy  of  an  operational  amplifier  depends  only  on  the  passive 
components  used  in  the  input  and  feedback  circuits  provided  that  there 
is  sufficient  negative  feedback  (usually  greater  than  60  db).  The  time 
that  is  required  for  the  amplifier  to  perform  a  calculation  is  an  inverse 
f miction  of  the  bandwidth  over  which  the  feedback  is  maintained. 
Thus  a  fundamental  problem  in  the  design  of  an  operational  amplifier 
is  the  development  of  sufficient  negative  feedback  over  a  reasonably 
broad  frequency  range.  The  associated  problem  is  the  realization  of 
satisfactory  stability  margins.  Finally  there  is  the  problem  of  reducing 
the  drift  which  is  inherent  in  direct  coupled  amplifiers  and  particularly 
troublesome  for  transistors  because  of  the  variation  in  their  character- 
istics with  temperature. 

The  first  step  in  the  design  is  the  blocking  out  of  the  configuration 
for  the  forward  gain  circuit  (designated  A  in  Fig.  1).  Three  primary  re- 
quirements must  be  satisfied: 

(1)  Stages  must  be  direct  coupled. 


TRANSISTOR    CIRCUITS    FOR   ANALOG    AND    DIGITAL    SYSTEMS 


297 


(2)  Amplifier  must  provide  one  net  phase  reversal. 

(3)  Amplifier  must  have  enough  current  gain  to  meet  accuracy  re- 
quirements. 

Three  possible  transistor  connections  are  available:  (a)  the  common 
base  connection  which  may  be  considered  analogous  to  the  common 
grid  vacuum  tube  connection;  (b)  the  common  emitter  connection 
which  is  analogous  to  the  common  cathode  connection;  and  (c)  the 
common  collector  connection  which  is  analogous  to  the  cathode  follower 
connection.  These  three  configurations  together  with  their  approximate 
equivalent  circuits  are  shown  in  Fig.  2.  It  has  been  shown^  that  for 
most  junction  transistors  the  circuit  element  a  is  given  by  the  expression 


a  =  sech 


W 


(1    +    PTrn) 


1/2 


(1)^ 


where  W  is  the  thickness  of  the  transistor  base  region,  Lm  is  the  diffusion 
length  and  t„,  the  lifetime  of  minority  charge  carriers  in  the  base  region, 


Rk 

I — ^AV 


E-        "J 

-^ Wvr 


Eo 


Rk     A/3EL  Rk  ^ 

•=0"   Rj    (i-A/i)"^       Rj  ^l 

(a)   MULTIPLICATION    BY   A 
CONSTANT   COEFFICIENT 


E, 


R. 


E  ^2 

E3  ^^ 


Rk 

I — vv\- 


Eo  =  E 


Rk    A/bEj 


p,  Rj    (i-A/3) 

(b)  ADDITION 


N  r-      . 

•RKEf: 


c 


§i — vw- 


£[Eo] 


A/3     £[el]        sl[eQ 


Eo 


^N^^^^?|^-PH«[EJ 


(d)  DIFFERENTIATION 


(l-A/3)    pRC     ~       pRC 
(C)    INTEGRATION 

note;       £[Eo]  =  LAPLACE    TRANSFORM    OF    OUTPUT  VOLTAGE 
£[Ei1  =  LAPLACE    TRANSFORM    OF    INPUT  VOLTAGE 

p  =  jco 
Fig.  1  —  Summary  of  operational  amplifiers. 


*  This  expression  assumes  that  the  injection  factor  y  and  the  collector  efficiency 
at  are  both  unity.  This  is  a  good  approximation  for  all  alloy  junction  transistors 
and  most  grown  junction  transistors. 


298  THE   BELL   SYSTEM   TECHNICAL  JOURNAL,   MARCH    1956 

and  7?  —  ju.  At  frequencies  less  than  Ua/^ir,  (1)  can  be  approximated  by 


a  = 


1  +  ^ 


(2)' 


COa 


where  ao  is  the  low  frequency  value  of 


a  ^  1 


l/TT 
2U. 


and 


2.4Z). 


CCa    = 


w 


(Dm  is  the  diffusion  constant  for  the  minority  charge  carriers  in  the  base 
region).  A  readily  measured  parameter  called  alpha  (a),  the  short 
circuit  current  gain  of  a  junction  transistor  in  the  common  base  connec- 


SCHEMATIC 


Zc   = 


EQUIVALENT   CIRCUIT 


s                               ^ — 

*■            e/ 

(V 

b 

=^ V\V 


aZcLe 


Tb 


(a)    COMMON    BASE 
Lb 


: 


rb 


Zed -a) 


aZc'Lb 

'X, 


■re 


(b)    COMMON    EMITTER 

i^b        rb 


aZc 


re 


aZcLb 


Zed -a) 


(C)    COMMON    COLLECTOR 


re 


1  +  prcCc 


a 

P 

— 

ao 

i-hP 

re  =  COLLECTOR    RESISTANCE 
Cc  =  COLLECTOR   CAPACITANCE 


ZTT 


ALPHA-CUTOFF  FREQUENCY 


Fig.  2  —  Basic  transistor  connections. 


TRANSISTOR   CIRCUITS   FOR   ANALOG   AND    DIGITAL   SYSTEMS         299 

tion,  is  related  to  a  by  the  equation 

aZe  -{■  n  /^x 

Ze  +  n 

For  most  junction  transistors  the  base  resistance,  n ,  is  much  smaller 
than  the  collector  impedance  |  Zc  |,  at  frequencies  less  than  Wa/27r.  There- 
fore, a  ^  a  and  Ua/^ir  is  very  nearly  equal  to  the  alpha-cutoff  frequency, 
the  frequency  at  which  |  a  |  is  down  by  3  db. 

The  transistor  parameters  r^  and  n  are  actually  frequency  sensitive 
and  should  be  represented  as  impedances.  However,  good  agreement 
between  theory  and  experiment  is  obtained  at  frequencies  less  than 
Wa/27r  with  re  and  n  assumed  constant. 

The  choice  of  an  appropriate  transistor  connection  for  a  direct  coupled, 
negative  feedback  amplifier,  is  based  on  the  following  reasoning.  The 
common  base  connection  may  be  ruled  out  immediately  because  this 
connection  does  not  provide  current  gain  unless  a  transformer  interstage 
is  used.  The  common  emitter  connection  provides  short  circuit  current 
gain  and  a  phase  reversal  for  each  stage.  Thus  if  the  amplifier  is  com- 
posed of  an  odd  number  of  common  emitter  stages,  all  three  requirements 
previously  listed,  are  satisfied.  A  common  emitter  cascade  has  the  addi- 
tional practical  advantage,  that  by  alternating  n-p-n  and  p-n-p  types  of 
transistors,  the  stages  can  be  direct  coupled  with  practically  zero  inter- 
stage  loss. 

The  common  collector  connection  provides  short  circuit  current  gain 
but  no  phase  reversal.  Consequently,  the  dc  amplifier  cannot  consist 
entirely  of  common  collector  stages  and  operate  as  a  negative  feedback 
amplifier.  This  paper  will  consider  only  the  common  emitter  connection 
since,  in  general,  for  the  same  number  of  transistor  stages,  the  common 
emitter  cascade  provides  more  current  gain  than  a  cascade  composed  of 
both  common  collector  and  common  emitter  stages. 

2.1  Evaluation  of  External  Voltage  Gain 

Since  the  equivalent  circuit  of  the  junction  transistor  is  current  acti- 
vated, it  is  convenient  to  treat  feedback  in  a  single  loop  transistor  ampli- 
fier as  a  loop  current  transmission  (refer  to  Appendix  I)  instead  of  as  a 
loop  voltage  transmission  which  is  commonly  used  for  single  loop  vacuum 
tube  amplifiers.^  Fig.  3  shows  a  single  loop  feedback  amplifier  in  which 
a  fraction  of  the  output  current  is  fed  back  to  the  input.  A  is  defined  as 
the  short  circuit  current  gain  of  the  amplifier  without  feedback,  and  jS  is 
defined  as  the  fraction  of  the  short  circuit  output  current  (or  Norton 


300 


THE    BELL    SYSTEM    TECHNICAL   JOURNAL,    MARCH    1956 


equivalent  circuit  current)  fed  back  to  the  input  summing  node.  With 
these  definitions, 

he  =  A/in'  (4) 


la  -  131. 


sc 


where  /sc  is  the  Norton  equivalent  short  circuit  current. 
From  Kirchhoff's  first  law 

/in    =  /in  +  Iff 
Combining  relations  (4)  to  (6)  yields 


'sc 


A 


(5) 


(6) 


(7) 


/in       1  -  A^ 
Expression  (7)  provides  a  convenient  method  for  evaluating  the  external 


^ 

[|N 

I  IN 

> 

^ 

Fig.  3  —  Single  loop  feedback  amplifier. 

voltage  gain  of  an  operational  amplifier.  Fig.  4  shows  a  generalized  op- 
erational amplifier  with  N  inputs.  With  this  configuration, 


IN 


j=i  L 


TTT  he     r,        I 


Zj 


(S) 


where  Ej ,  j  =   1,2,  •  ■  •  ,  N,  are  the  N  input  voltages  referred  to  the 
ground  node. 
Zj,j—  1,2,  •  ■  ■  ,  N,  are  the  A^  input  impedances 
ZiN    is  the  input  impedance   of  the  amplifier  measured  at  the 
summing  node  with  the  feedback  loop  opened. 


Eo 


//i 


sc 


UT 


'IN 


la  = 


A 


Eovr  = 


Zk 

/sc  ~  / 


(3 


Rl        Zovt 


(5>) 


(10) 


TRANSISTOR   CIRCUITS   FOR   ANALOG   AND    DIGITAL   SYSTEMS         301 


where  Zovr  is  the  output  impedance  of  the  amplifier  measured  with  the 
feedback  loop  opened.  The  expression  for  the  output  voltage  is  obtained 
by  combining  (7),  (8),  (9),  and  (10). 


E, 


OUT 


N  y 

=   zL  ^i  7" 

;=1  ^i 


A^  + 


3  =  1       ^1    _ 


(iir 


where 


A^  =  A 


1  - 


'IN 


\Ri 


+ 


/OUT 


1    _^    ^    +        ^^^ 

Rl        Zovr 


IA/3  is  equal  to  the  current  returned  to  the  summing  node  when  a  unit 


Ei 


Z, 


MN 


1/3  Zk 


I  IN 


Zn 


Zls 


J7 


1/5  Zk 


Equt 


NORTON    EQUIVALENT    CIRCUIT 


Fig.  4  —  Generalized  operational  amplifier. 

icurrent  is  placed  into  the  base  of  the  first  transistor  stage  (/in    =   1). 
If  I  A^  1  is  much  greater  than  ]  Zj^'/Zr  \  and 


1  +  L 


'IN 


then 


N 


Eqvt   —     ~  2^  J^j  nT 


(12) 


y=i 


The  accuracy  of  the  operational  amplifier  depends  on  the  magnitude  of 
AjS  and  the  precision  of  the  components  used  in  the  input  and  feedback 
networks  as  can  be  seen  from  (11).  There  is  negligible  interaction  between 
the  input  voltages  because  the  input  impedance  at  the  summing  node  is 
equal  to  Zin'  divided  by  (1  —  A^)?  This  impedance  is  usually  negligibly 
tsmall  compared  to  the  impedances  used  in  the  input  circuit. 

*  In  general,  E,-  and  Eout  are  the  Laplace  transforms  of  the  input  and  output 
fvoltages,  respectively. 


302 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    1956 


2.2.  Methods  Used  to  Shape  the  Loop  Current  Transmission 

An  essential  consideration  in  the  design  of  a  feedback  amplifier  is  the 
provision  of  adequate  margins  against  instability.  In  order  to  accomplish 
this  objective,  it  is  necessary  to  choose  a  criterion  of  stability.  In  Ap- 
pendix I  it  is  shown  that  it  is  convenient  and  valid  to  base  the  stability 
of  single  loop  transistor  feedback  amplifiers  on  the  loop  current  trans- 
mission. In  order  to  calculate  the  loop  current  transmission  of  the  dc 
amplifier,  the  feedback  loop  is  opened  at  a  convenient  point  in  the  cir- 
cuit, usually  at  the  base  of  one  of  the  transistors,  and  a  unit  current  is 
injected  into  the  base  (refer  to  Fig.  24).  The  other  side  of  the  opened 
loop  is  connected  to  ground  through  a  resistance  (r^  -j-  r^)  and  voltage 
Veli  •  In  many  instances,  the  voltage  re/4  can  be  neglected.  If  |  Zj?  |  and 


3=1  Zj 


I 


are  much  greater  than  |  Z 


IN 


then  A/3  is  very  nearly  equal  to  the  loop 
ciu'rent  transmission.  For  absolute  stability^  the  amplitude  of  the  loop 
current  transmission  must  be  less  than  unity  before  the  phase  shift 
(from  the  low  frequency  value)  exceeds  180°.  Consequently,  this  charac- 
teristic must  be  controlled  or  properly  shaped  over  a  wide  frequency 


10 

_J 

LU 

O 
LU 
Q 


< 

o 

\- 
z 

LU 

a. 
o 


40 

U), 

a;,' 

Wa 

^{\-\-S)u)^ 

\ 

ao 

^" 

■\ 

\, 

\J 

i 

"^ 

\ 

30 

1- 

ao+cT 

t-  — 

.      ao 

\ 

7~' 

\ 

\ 

^ 

AM  PL 

ITUC 

E 

\ 

\ 

20 
10 

i-ao 

+  7_ 

AMPLITUDE' 

(WITH    LOCAL 

FEEDBACK) 

\ 

\ 

\ 
\ 

S 

< 

PHASE   (WITH 
LOCAL  FEEDBACK) 

phase\ 

\ 

y 

\ 
> 

\ 

\ 

ao 

0 

^'^ 

f' 

\+S       1 

-270° 

X 

— . 

s 
\ 

\ 

N 

d 

«/c 

-10 
20 
30 
40 

"^ 

N 

\ 

^ 

^ 

\ 

^ 

\ 

\ 

•180 


-200 


-220 


•240 


•260  uj 

_l 

2 
< 


-280 


•300 


LU 

<     I 

I 

Q- 


-320 


■340 


,02      2  5       ,q3      2  ^       ,o4      •=;  S       ,q5      t  S       ,q6 


5   ■/^4   2     5   ,„s   2     5   ,„6   2     5   ^q7   2     5   jq8 
FREQUENCY  IN  CYCLES  PER  SECOND 


Fig.  5  —  Current  transmission  of  a  common  emitter  stage. 


TRANSISTOR   CIRCUITS   FOR   ANALOG   AND   DIGITAL   SYSTEMS         303 

band.  In  addition,  it  is  desirable  that  the  feedback  fall  off  at  a  rate  equal 
to  or  less  than  9  db  per  octave  in  order  to  insure  that  the  dc  aniplifier 
has  a  satisfactory  transient  response. 

Three  methods  of  shaping  are  described  in  this  paper;  local  feedback 
shaping,  interstage  network  shaping,  and  (3  circuit  shaping.  Local  feed- 
back shaping  will  be  described  first.  The  analysis  starts  by  considering 
the  current  transmission  of  a  common  emitter  stage,  ecjuivalent  circuit 
shown  in  Fig.  2(b).  If  the  stage  operates  into  a  load  resistance  Rl  ,  then 
to  a  good  approximation  the  current  transmission  is  given  by 


where 


Gr  =  r"  =  ^  ~  ^°  +/ (13)^ 

^^    1  +  ^  + '^ 

wi        a)aCOc(l  —  ao  -\-  8) 

RL+Te 


8    = 


COl    = 


(1  -  ao  +  8) 
1  +  5,1 


-^  ^  alpha-cutoff  frequency 
Ztt 


1 


Uc 


(7?x,  +  re)Cc 


It  is  apparent  from  expression  (13)  that  if  (1  —  Oo  +  8)  is  less  than  0.1, 
then  the  current  gain  of  the  common  emitter  stage  falls  off  at  a  rate  of 
6  db  per  octave  with  a  corner  frequency  at  wi  .f  A  second  6  db  per  octave 
cutoff  with  a  corner  frequency  at  [co^  +  (1  +  5)aJc]  is  introduced  by  the 
p"  term  in  the  denominator  of  (13).  A  typical  transmission  characteristic 
is  shown  in  Fig.  .5.  The  current  gain  of  the  common  emitter  stage  is  unity 
at  a  frequency  equal  to 

ao 


1  +5    I     1 


*  Expressions   (13)   and   (14)   are  poor  approximations  at   frequencies  above 

'    coo/27r. 

'        t  Strictly  speaking  the  corner  frequency  is  equal  to  01/2 tt.  However,  for  sim- 
plicity, corner  frequencies  will  be  expressed  as  radian  frequencies. 


304  THE    BELL   SYSTEM    TECHNICAL   JOURNAL,    MARCH    1956 

Since  the  phase  crossover  of  A|S*  is  usually  placed  below  this  frequency, 
the  principal  effect  of  the  second  cutoff  is  to  introduce  excess  phase.  This 
excess  phase  can  be  minimized  by  operating  the  stage  into  the  smallest 
load  resistance  possible,  thus  maximizing  Wc .  j 

An  undesirable  property  of  the  common  emitter  transmission  charac- 
teristic is  that  the  corner  frequency  coi  occurs  at  a  relatively  low  fre- 
quency. However,  the  corner  frequency  can  be  increased  by  using  local 
feedback  as  shown  in  Fig.  6(a).  Shunt  feedback  is  used  in  order  to  pro- 
vide a  low  input  impedance  for  the  preceding  stage  to  operate  into.  The 
amplitude  and  phase  of  the  current  transmission  is  controlled  prin- 
cipally by  the  impedances  Z\  and  Z2  .  If  |  A&  \  is  much  greater  than  one, 
and  if  /3  ;^  ^1/^2  ,  then  from  (7)  the  current  transmission  of  the  stage  is 
approximately  equal  to  —  Z2/Z1  .  Because  of  the  relatively  small  size  of 
A^  for  a  single  stage,  this  approximation  is  only  valid  for  a  very  limited 
range  of  values  of  Zi  and  Z2  .  If  Zi  and  Zi  are  represented  as  resistances 
R\  and  Ri  ,  then  the  current  transmission  of  the  circuit  is  given  to  a  good 
approximation  by 


tto 


h.  _  R2         1  —  gp  +  7 

^^  =  /i=  ~{R2  +  n)r_^p_^  v' 


where 


7    = 


coi    = 


COc    = 


Co/  COaCOcCl     —    Oo    +    7), 

R\    +    Te  _,Rl   +   Te 

R2   +   ^6 


I 

(14) 


(/?2   + 

rb)rc 

i22  +  n 

(1  +ao 

+  ro 
+  7) 

1  +  7 

1 

{R,  -f  re)Cc  i 


By  comparing  (14)  with  (13),  it  is  evident  that  the  negative  feedback 
has  reduced  the  low-frequency  current  gain  from  ao/(l  —  ao)  (5  may 
usually  be  neglected)  to 


( 


R2         \  I  «0  \   _ ,  ^2 


R2  +  rj  \1  -  ao  +  7/       ^1  +  re 


(if  7  >  1  -  ao) 


.-•! 


*  The  phase  crossover  of  A/3  is  equal  to  the  frequency  at  which  the  phase  shift 
of  A/3  from  its  low-frequency  value  is  180°. 


I 


TRANSISTOR   CIRCUITS    FOR    ANALOG    AND    DIGITAL   SYSTEMS         305 

The  half  power  frequency,  however,  has  been  increased  from 

1—   Oo  ,  1—   Oo  +  7 

t:^        1  +  7 ,  1 

as  shown  by  the  dashed  curves  in  Fig.  5.* 

The  bandwidth  of  the  common  emitter  stage  can  be  increased  without 
reducing  the  current  gain  at  dc  and  low-frequencies  by  representing  Zi 
by  a  resistance  Ri ,  and  Z2  by  a  resistance  R2  in  series  with  a  condenser 
C2 .  If  I/R2C2  is  much  smaller  than  co/,  then  the  current  transmission  of 
the  stage  is  given  by  (14)  multiplied  by  the  factor 


P 


1  + 


C04 
P 


(15) 


where 


602 


Wi 


H^^i 


1  -  cro  + 


Ri  +  re 


C2(/?2  +  r6)(l  -  ao  +  7) 


The  current  transmission  for  this  case  is  plotted  in  Fig.  6(b).  The  con- 
denser d  introduces  a  rising  6  db  per  octave  asymptote  with  a  corner 
frequency  at  wi .  At  dc  the  current  gain  is  equal  to 


ao 


1  —  ao  +  5 


A  second  method  of  shaping  the  loop  current  transmission  char- 
acteristic of  a  feedback  amplifier  is  by  means  of  interstage  networks. 
These  networks  are  usually  used  for  reducing  the  loop  current  gain  at 
relatively  low  frequencies  while  introducing  negligible  phase  lag  near 
the  gainf  and  phase  crossover  frequencies.  Interstage  networks  should 
be  designed  to  take  advantage  of  the  variable  transistor  input  impedance. 
The  input  impedance  of  a  transistor  in  the  common  emitter  connection 


*  In  Figs.  5  and  6(b),  the  factor  R^/iRi  +  n)  is  assumed  equal  to  unity.  This  is 
'  a  good  approximation  since  in  practice  R2  is  equal  to  several  thousand  ohms  while 
rt  is  equal  to  about  100  ohms. 

t  The  gain  crossover  frequency  is  equal  to  the  frequency  at  which  the  magni- 
tude of  Al3  is  unity. 


306 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    1956 


is  given  by  the  expression 


'INP 


UT    =    ?'6  +   ^e(l    —    Gi) 


(16)1 


where  Gj  is  the  current  transmission  given  by  (13).  If  Gi  at  dc  is  much' 
greater  than  1,  then  the  input  impedance  and  the  current  transmission 
of  the  common  emitter  stage  fall  off  at  about  the  same  rate  and  with 
approximately  the  same  corner  frequency  (wi).  The  input  impedance 
finally  reaches  a  limiting  value  equal  to  r^  +  Vb . 

A  particularly  useful  interstage  network  is  shown  in  Fig.  7(a).  This 
network  is  analyzed  in  Appendix  II  and  Fig.  7(b)  shoAvs  a  plot  of  the 


60 


50 


40 


30 


20 


Z 
< 


z 

UJ 


10 


tr      0 
cr 

D 
U 

-10 


-20 


-30 


(a: 


EQUIVALENT    CIRCUIT 


\ 

\ 

(b) 

\ 

AMPLITUDE 

an 

\ 

(WITHOUT    LOCAL 

\ 

FEEDBACK) 

1-ao+d" 

^   - 

^" 

\ 

■*•> 

1 

^ 

s 

.AMPLITUDE 

^4 

^>CiL 

^'' 

,^_ 

•—  ^  ■ 

■~^ 

r"**^ 

cvz 

r^ 

i 

/ 

"^v 

^ 

'               1 

>^ 

/ 

V 

\ 
\ 
\ 

X 

V 

^  ao 

i-ao+ 

7     - 

/ 

A 

/  ^ 

\ 

s. 

\ 

k 

\ 

\ 

\ 

/ 

/ 

PH/ 

>s> 

/ 

\ 

\ 
\ 

\ 
V 

\ 

\ 

PHASE            N 

\ 

(WITHOUT    LOCAL 

\ 

s 

FEEDBACK) 

s 

w 

k. 

- 

^-. 

■"••^^, 

120 


140 


-160  10 

UJ 

m 
cr 

-180  liJ 

Q 
Z 

-200  ^ 

z 
< 


-220  , 


10 


2  5p2  5,2  5.2  5,2  5 

-!•=  in^  m^  in5 


10'= 


lO-^*  10^  10= 

FREQUENCY  IN  CYCLES  PER  SECOND 


lO'' 


-240  ' 


260 


-  -280 


10' 


Fig.  6  —  Negative  feedback  applied  to  a  common  emitter  stage. 


TRANSISTOR   CIRCUITS   FOR   ANALOG   AND   DIGITAL   SYSTEMS 


307 


resulting  current  transmission.  The  amplitude  of  the  transmission  falls 
off  at  a  rate  of  6  db  per  octave  with  the  corner  frequency  C05  determined 
by  C'3  and  the  low  frequency  value  of  the  transistor  input  impedance. 
The  inductance  L3  introduces  a  12  db  per  octave  rising  asymptote  with 
a  corner  frequency  at  C03  =  WLsCs  .  The  corner  frequencies  C03  and  C05 
are  selected  in  order  to  obtain  a  desirable  loop  current  transmission 
characteristic  (specific  transmission  characteristics  are  presented  in  Sec- 
tions 3.0  and  4.0).  The  half  power  frequency  of  the  current  transmission 
of  the  transistor,  wi ,  does  not.  appear  directly  in  the  transmission  char- 
acteristic of  the  circuit  because  of  the  variation  in  the  transistor  input 
impedance  with  frequency. 
The  overall  (3  circuit  of  the  feedback  amplifier  can  also  be  used  for 


i-ao+(J 


I  ^ 
s 

LU 
Q 

z 
I     - 


<l< 

z 
< 

15 


Z 
UI 

cc 

D 

u 

Q 
UJ 

y 

< 

2 

a 
o 

z 


40 


20 


-20 


-40 


-60 


-80 


(b) 

/ 

/ 

CU5 

u 

■^3/ 

y' 

* 

^^^ 

A^ 

^ 

^s^ 

1 

\. 

^N, 

/ 

^s^ 

S^ 

/ 

\ 

\ 

AMPLITUDE 

/ 
/ 

\ 

\^ 

/ 

\ 

\ 
\ 
\ 

\ 

> 

X 

X 

V 

1 

1 
/ 
/ 
/ 

1 
1 
1 
1 

1 

_ 

\ 

\ 

\ 

\ 

\ 

s,. 

/ 

cu,(rb+ 

'   Te 

\ 

\-do+l 

W 

\ 

.PHASE 

^ 

Tb+le-l-Ra-K^iLa 

^^ 

/ 

\. 

*^.., 



— 

^** 

X 

— - 

N 

- 

-135 


10 


LU 
UJ 

isog 


z 
< 

-225  1}^ 


< 

I 
a. 


-270 


102        "  "^       \0^        "  =       10^        -^  =       105 

FREQUENCY    IN    CYCLES    PER    SECOND 


Fig.  7  —  Interstage  shaping  network. 


lO'' 


308  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 

shaping  the  loop  current  transmission.  If  the  feedback  impedance  Zk 
(Fig.  4)  consists  of  a  resistance  Rk  and  condenser  Ck  in  parallel,  then 
the  loop  current  transmission  is  modified  by  the  factor 


1  + 


CO; 


1  +  ^ 


COS 


(17) 


where 


C07  = 


C08  = 


RkCk 

(Rl_±_Rk) 

RlRkC  K 


Since  Zk  affects  the  external  voltage  gain  of  the  operational  amplifier, 
(11),  the  corner  frequency  C07  must  be  located  outside  of  the  useful  fre- 
quency band.  Usually  it  is  placed  near  the  gain  crossover  frequency  in 
order  to  improve  the  phase  margin  and  the  transient  response  of  the 
amplifier. 

In  Sections  3.0  and  4.0,  the  above  shaping  techniques  are  used  in  the 
design  of  specific  operational  amplifiers. 

3.0.  THE   SUMMING   AMPLIFIER 

3.1.  Circuit  Arrangement 

The  schematic  diagram  of  a  dc  summing  amplifier  is  shown  in  Fig.  8. 
From  the  discussion  in  Section  2.0  it  is  apparent  that  each  common 
emitter  stage  will  contribute  more  than  90  degrees  of  high-frequency 
phase  lag.  Consequently,  while  the  magnitude  of  the  low-frequency  : 
feedback  increases  with  the  number  of  stages,  this  is  at  the  expense  of  , 
the  bandwidth  over  which  the  negative  feedback  can  be  maintained. 
It  is  possible  to  develop  80  db  of  negative  feedback  at  dc  with  three 
common  emitter  stages.  This  corresponds  to  a  dc  accuracy  of  one  part 
in  10,000.  In  addition,  the  feedback  can  be  maintained  over  a  broad 
enough  band  in  order  to  permit  full  accuracy  to  be  attained  in  about 
100  microseconds.  Thus  it  is  evident  that  the  choice  of  three  stages  repre- 
sents a  satisfactory  compromise  between  accuracy  and  bandwidth  ob- 
jectives. 

The  output  stage  of  the  amplifier  is  designed  for  a  maximum  power 
dissipation  of  75  milliwats  and  maximum  voltage  swing  of  ±25  volts 


I 


TRANSISTOR   CIRCUITS   FOR   ANALOG   AND    DIGITAL   SYSTEMS 


309 


when  operating  into  an  external  load  resistance  equal  to  or  greater  than 
50,000  ohms.  A  p-n-p  transistor  is  used  in  the  second  stage  and  n-p-n 
transistors  are  used  in  the  first  and  third  stages.  This  circuit  arrangement 
makes  it  possible  to  connect  the  collector  of  one  transistor  directly  to 
the  base  of  the  following  transistor  without  introducing  appreciable 
interstage  loss.  ''Shot"  noise"  and  dc  drift  are  minimized  by  operating 
the  first  stage  at  the  relatively  low  collector  current  of  0.25  milliamperes. 
The  110,000-ohm  resistor  provides  the  collector  current  for  the  first 
stage,  and  the  4,700-ohm  resistor  provides  3.8  milliamperes  of  collector 
current  for  the  second  stage.  The  series  6,800-ohm  resistor  between  the 
xcond  and  third  stages,  reduces  the  collector  to  emitter  potential  of  the 
second  stage  to  about  4.5  volts. 

The  loop  current  transmission  is  shaped  by  use  of  local  feedback  ap- 
plied to  the  second  stage,  by  an  interstage  network  connected  between 
the  second  and  third  stages,  and  by  the  overall  (3  circuit.  The  200-ohm 
resistor  in  the  collector  circuit  of  the  second  stage  is,  with  reference  to 
Fig.  6(a),  Zi  .  The  impedance  of  the  interstage  network  can  be  neglected 
since  it  is  small  compared  to  200  ohms  at  all  frequencies  for  which  the 
local  feedback  is  effective.  The  interstage  network  is  connected  between 
the  second  and  third  stages  in  order  to  minimize  the  output  noise  voltage. 
^^'ith  this  circuit  arrangement,  practically  all  of  the  output  noise  voltage 


iE 


250  K 


IN 


+  33V 


5MUf 

Hf- 


20on 


n-p-n 


250  K 
2.4  K  200  n 


0.01/U.F 


p-n-p 


■llOK 


100  K    POT. 

MANUAL 

ZERO    SET 


I 

+  33V 


I 

+  4.5V 


OUT 


5>UH 


-45V  -27V     +33V 


Fig.  8  — ■  DC  summing  amplifier. 


310  THE   BELL   SYSTEM   TECHNICAL  JOURNAL,   MARCH    1956 


120 


100 


UJ 

U     80 

LU 

o 

z 
"    60 

< 


< 

IS 


LU 

a: 
tr 

D 
U 

Q. 

o 
o 

_) 


40 


20 


■20 


-40 


,-' 

../>' 

--T 

.-' 

— 

364 

LOCAL    _ 
FEEDBACK"^- 

^-' 

.-1 

41,000 

-^ 

^ 

d      6 

630 

\, 

.     12 

N 
\ 

?,000 

--.., 

s^-> 

^-. 

'S 

'-.. 

\ 

\ 

V 
\ 
\ 
V 

2ND      •^^ 

STAGE          ^ 

s 

\ 

-.. 

"-. 
^-. 

1ST    &    3RD 
STAGES 

N 

\ 

0.5/ZF 

^■-S;:-.-, 

^ 

\, 

\ 
\ 

> 

\ 

10' 


10-^ 


10- 


10' 


10' 


FRFOUENCY     IN    CYCLES    PER    SECOND 

Fig.  9  —  Gain-frequency  asymptotes  for  summing  amplifier. 

is  generated  in  the  first  transistor  stage.  If  the  transistor  in  the  first 
stage  has  a  noise  figure  less  than  10  db  at  1,000  cycles  per  second,  then 
the  RMS  output  noise  voltage  is  less  than  0.5  millivolts. 

Fig.  9  shows  a  plot  of  the  gain-frequency  asymptotes  for  the  sum- 
ming amplifier  determined  from  (13),  (14),  (15),  (17),  and   (A6)  under 
the  assumption  that  the  alphas  and  alpha-cutoff  frequencies  of  the  tran- 
sistors are  0.985  and  3  mc,  respectively.  The  corner  frequencies  intro-' 
duced  by  the  0.5  microfarad  condenser  in  the  interstage  network,  thel 
local  feedback  circuit,  and  the  cutoff  of  the  first  and  third  stages  are  so 
located  that  the  current  transmission  falls  off  at  an  initial  rate  of  about' 
9  db  per  octave.  This  slope  is  joined  to  the  final  asymptote  of  the  loop 
transmission  by  means  of  a  step-type  of  transition.^   The  transition  is 
provided  by  3  rising  asymptotes  due  to  the  interstage  shaping  network, 
and  the  overall  /S  circuit.  An  especially  large  phase  margin  is  used  in  order 
to  insure  a  good  transient  performance. 

Fig.  10  shows  the  amplitude  and  phase  of  the  loop  current  trans- 
mission. When  the  amplitude  of  the  transmission  is  0  db,  the  phase  angle 
is  -292°,  and  when  the  phase  angle  is  —360°,  the  amplitude  is  27.5  db 


TRANSISTOR   CIRCUITS   FOR  ANALOG   AND    DIGITAL   SYSTEMS 


311 


100 


LU 

m 
u 

LU 
Q 


<^ 

z 
< 

H 
Z 
UJ 

a. 
a. 

D 

o 

Q. 
O  " 

o 

_l 


80 


60 


40 


20 


20 


-40 


— 

■~-^ 

^•"v 

> 

\ 

^ 

AM 

PLITL 

DE 

\ 

1 

\ 

\ 

\ 

s 

> 

\. 

._ 

s 

>^ 

^r 

•—'' 

y^       "N   phase 

'PHASE     'nCROSSOVER 

s 

\ 

\ 

V 

/  GAIN^-N^ 
CROSSOVER 

N 

-27.5  DB 
95  =  -360° 

sv 

■160 


-200 


to 

LU 
-240  ^ 

O 

LU 

Q 

-280   7 


•320 


-360 


-400 


■440 


10= 
FREQUENCY  IN  CYCLES  PER  SECOND 


10^ 


10' 


Fig.  10  —  Loop  current  transmission  of  the  summing  amplifier. 

below  0  db.  The  amplifier  has  a  68°  phase  margin  and  27.5  db  gain  margin. 
In  order  to  insure  sufficient  feedback  at  dc  and  adequate  margins  against 
instability,  the  transistors  used  in  the  amplifier  should  have  alphas  in 
the  range  0.98  to  0.99  and  alpha-cutoff  frequencies  equal  to  or  greater 
than  2.5  mc. 


3.2.  Automatic  Zero  Set  of  the  dc  Summing  Amplifier 

The  application  of  germanium  junction  transistors  to  dc  amplifiers 
does  not  eliminate  the  problem  of  drift  normally  encountered  in  vacuum 
tube  circuits.  In  fact,  drift  is  more  severe  due  principally  to  the  varia- 
tion of  the  transistor  parameters  alpha  and  saturation  current  with 
temperature  variation.  Even  though  the  amplifier  has  80  db  of  negative 
feedback  at  dc,  this  feedback  does  not  eliminate  the  drift  introduced  by 
[the  first  transistor  stage.  Because  of  the  large  amount  of  dc  feedback, 
the  collector  current  of  the  first  stage  is  maintained  relatively  constant. 
The  collector  current  of  the  transistor  is  related  to  the  base  current  by 
the  equation 


Ic   = 


/c 


+ 


a 


I  —  a       1  —  a 


(18) 


[The  saturation  current,  Ico  ,  of  a  germanium  junction  transistor  doubles 
(approximately  for  every  11°C  increase  in  temperature.  The  factor 
a/(l  —  a)  increases  by  as  much  as  6  db  for  a  25°C  increase  in  tempera- 


312  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    1956 

ture.  Consequently,  the  base  current  of  the  first  stage,  Ih ,  and  the  output 
voltage  of  the  amplifier  must  change  with  temperature  in  order  to  main- ' 
tain  Ic  constant.  The  drift  due  to  the  temperature  variation  in  a  can  be 
reduced  by  operating  the  first  stage  at  a  low  value  of  collector  current. 
With  a  germanium  junction  transistor  in  the  first  stage  operating  at  a 
collector  current  of  0.25  milliamperes,  the  output  voltage  of  the  amplifier 
drifts  about  ±1.5  volts  over  a  temperature  range  of  0°C  to  50°C.  It  is 
possible  to  reduce  the  dc  drift  by  using  temperature  sensitive  elements 
in  the  amplifier.  •  In  general,  temperature  compensation  of  a  transistor 
dc  amplifier  requires  careful  selection  of  transistors  and  critical  adjust- 
ment of  the  dc  biases.  However,  even  with  the  best  adjustments,  tem- 
perature compensation  cannot  reduce  the  drift  in  the  amplifier  to  within 
typical  limits  such  as  ±5  millivolts  throughout  a  temperature  range  of  i 
0  to  50°C.  In  order  to  obtain  the  desired  accuracy  it  is  necessary  to  use 
an  automatic  zero  set  (AZS)  circuit.  t 

Fig.  11  shows  a  dc  summing  amplifier  and  a  circuit  arrangement  fori 
reducing  any  dc  drift  that  may  appear  at  the  output  of  the  amplifier. 
The  output  voltage  is  equal  to  the  negative  of  the  sum  of  the  input  volt- 
ages, where  each  input  voltage  is  multiplied  by  the  ratio  of  the  feedback 
resistor  to  its  input  resistor.  In  addition,  an  undesirable  dc  drift  voltage  ^ 
is  also  present  in  the  ovitput  voltage.  The  total  output  voltage  is 

^o.t   =    -i:^y|^  +  Adrift  (1!))^ 

In  order  to  isolate  the  drift  voltage,  the  A^  input  voltages  and  the  output 
voltage  are  applied  to  a  resistance  summing  network  composed  of  re- 
sistors Ro ,  Ri  ,  R2 ,  •  •  •  ,  Rn  ■  The  voltage  across  Rs  is  equal  to 

Es=^  Adrift  (20) 

if 

R,«Ro,R/;        j  =  1,2,  ■■'  ,N 
and 

RoRj  =  RkR,';       j  =  1,2,  ■■■  ,N 

The  voltage  E,  is  amplified  in  a  relatively  drift-free  narrow  band  dc 
amplifier  and  is  returned  as  a  drift  correcting  voltage  to  the  input  of  the 
dc  summing  amplifier.  If  the  gain  of  the  AZS  circuit  is  large,  the  drift 
voltage  at  the  output  of  the  summing  amplifier  can  be  made  very  small. 
Fig.  12  shows  the  circuit  diagram  of  a  summing  amplifier  which  uses 
a  mechanical  chopper  in  the  AZS  circuit.^^  The  AZS  circuit  consists  of  a 


TRANSISTOR   CIRCUITS    FOR   ANALOG   AND    DIGITAL   SYSTEMS         313 

resistance  summing  network,  a  400-cycle  synchronous  chopper,  and  a 
tuned  400-cycle  amplifier.  Any  drift  in  the  summing  amplifier  will  pro- 
duce a  dc  voltage  Es  at  the  output  of  the  summing  network.  The  chopper 
converts  the  dc  voltage  into  a  400  cycles  per  second  waveform.  The 
fundamental  frequency  in  the  waveform  is  amplified  by  a  factor  of  about 
400,000  by  the  tuned  amplifier.  The  synchronous  chopper  rectifies  the 
sinusoidal  output  voltage  and  preserves  the  original  dc  polarity  of  Eg  . 
The  rectified  voltage  is  filtered  and  fed  back  to  the  summing  amplifier 
as  an  additional  input  current.  The  loop  voltage  gain  of  the  AZS  circuit 
at  dc  is  about  54  db.  Any  dc  or  low-frequency  drift  in  the  summing 
amplifier  is  reduced  by  a  factor  of  about  500  by  the  AZS  circuit.  The 
drift  throughout  a  temperature  range  of  0  to  50°C  is  reduced  to  ±3 
millivolts. 

Since  the  drift  in  the  summing  amplifier  changes  at  a  relatively  slow 
rate,  the  loop  voltage  gain  of  the  AZS  circuit  can  be  cutoff  at  a  relatively 
low  frequency.  In  this  particular  case  the  loop  voltage  gain  is  zero  db  at 
about  10  cycles  per  second. 


4.0.  THE   INTEGRATOR 

4.1.  Basic  Design  Considerations 

The  design  principles  previously  discussed  are  illustrated  in  this  sec- 
tion by  the  design  of  a  transistor  integrator  for  application  in  a  voltage 


VvV 


-OUT 


Fig.  11  —  DC  summing  amplifier  with  automatic  zero  set. 


314  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    1956 


TRANSISTOR   CIRCUITS    FOR   ANALOG   AND    DIGITAL   SYSTEMS         315 


encoder.  The  integrator  is  required  to  generate  a  15-volt  ramp  which  is 
linear  and  has  a  constant  slope  to  within  one  part  in  8,000.  This  ramp  is 
to  have  a  slope  of  5  millivolts  per  microsecond  for  an  interval  of  3,000 
microseconds. 

The  first  step  in  the  design  is  to  determine  the  bandwidth  over  which 
the  negative  feedback  must  be  maintained  in  order  to  realize  the  desired 
output  voltage  linearity.  The  relationship  between  the  output  and  input 
voltage  of  the  integrator  can  be  obtained  from  expression  (11)  by  sub- 
stituting (1/pc)  for  Zk  and  R  for  Zj  (refer  to  Fig.  1). 


£l-C'outJ   — 


pRC 


A/3  +  Zr^'pC 


1  -  AjS  + 


-nN_ 
R 


(21) 


where  ce[£'ouT]  and  JSiii'iN]  are  the  Laplace  transforms  of  the  output  and 
input  voltages,  respectively.  In  order  to  generate  the  voltage  ramp,  a 
step  voltage  of  amplitude  E  is  applied  to  the  input  of  the  integrator.  The 
term  Zy^  jR  is  negligible  compared  to  unity  at  all  frequencies.  Therefore, 


£L-£'outJ  — 


E     \     A& 


+ 


EZ 


IN 


1 


'^-RC  Ll  -  A&\         pR    \\  -  A^_ 
It  will  be  assumed  that  A/3  is  given  by  the  expression 

-K 


(22) 


A^  = 


V 


)0  +  ^T 


(i  +  -M(i  +  ^ 


(23) 


Expression  (23)  implies  that  A/3  falls  off  at  a  rate  of  6  db  per  octave  at 
low  frequencies  and  12  db  per  octave  at  high  frequencies.  The  output 
\  voltage  of  the  integrator,  as  a  function  of  time,  is  readily  evaluated  by 
substituting  (23)  into  (22)  and  taking  the  inverse  Laplace  transform  of 
the  results.  A  good  approximation  for  the  output  voltage  is 


^OUT    — 


E 


RC 


+ 


2K 


^-[(2w2+«l)(/2]    ^;„    -x/W 


sm 


Vk> 


OJo 


■iC02M 


ER 


(24)^ 


IN 


R 


[1  _  e-(-i'W  _!_  g-[(2<-2+.i)t/2i  ^Qg  ^Tkc.,!] 


The  linear  voltage  ramp  is  expressed  by  the  term  —  (Et/RC) .  The 
additional  terms  introduce  nonlinearities.  The  voltage  ramp  has  a  slope 
of  5  millivolts  per  microsecond  for  E  =   —21  volts,  R  =  42,000  ohms, 

*  In  evaluating  jE'out  it  was  assumed  that  Zm'  was  equal  to  a  fixed  resistance 
Rin' ,  the  low  frequency  input  resistance  to  the  first  common  emitter  stage.  A 
complete  analysis  indicates  that  this  assumption  makes  the  design  conservative. 


31G 


THE    BELL   SYSTEM    TECHNICAL   JOURNAL,    MARCH    1956 


and  C  =  0.1  microfarads.  For  these  circuit  values,  and  K  =  10,000 
(corresponding  to  80  db  of  feedback)  the  nonhnear  terms  are  less  than 
1/8,000  of  the  linear  term  (evaluated  when  /  =  4  X  10"^  seconds)  if 
/i  ^  30  cycles  per  second,  J2  ^  800  cycles  per  second,  and  if  the  first 
1000  microseconds  of  the  voltage  ramp  are  not  used.  Consequently,  80 
db  of  negative  feedback  must  be  maintained  over  a  band  extending  from 
30  to  800  cycles  per  second  in  order  to  realize  the  desired  output  voltage 
linearity. 

4.2.  Detailed  Circuit  Arrangement 

Fig.  13  shows  the  circuit  diagram  of  the  integrator.  The  method  of 
biasing  is  the  same  as  is  used  in  the  summing  amplifier.  The  200,000-ohm 
resistor  provides  approximately  0.5  milliamperes  of  collector  current  for 
the  first  stage.  The  40,000-ohm  resistor  provides  approximately  0.9 
milliamperes  of  collector  current  for  the  second  stage.  The  output  stage 
is  designed  for  a  maximum  power  dissipation  of  120  milliwatts  and  for 
an  output  voltage  swing  between  —5  and  +24  volts  when  operating 
into  a  load  resistance  equal  to  or  greater  than  40,000  ohms. 


J+'08V 


•  +  108V 


42  K 


D2 

44- 


C 


0.01>(/F  o.l/iF 

2.4K 


270  K 


I 

+  I08V 


1MEG 


200n 

\ — vw 

2>U.F 


200  K 


rVWA/^An 

j  100  K       [ 


POT.        I 


I 


OUT 


-10.5V 


+  108V 


+  4.5V 


•45V  -10.5V 


Fig.  13  —  Integrator. 


TRANSISTOR    CIRCUITS    FOR   ANALOG    AND    DIGITAL   SYSTEMS         317 


1   !!3 

{       LU 

I  5 
u 
ai 
a 


1     z 
< 


140 
120 

N 

100 

.^^ 

\ 

AMPLITUDE 

v. 

80 

^""^ 

\ 

\ 

\ 

\ 

N 

\, 

60 

\ 
\ 

> 

\ 

\ 

\ 

s 

40 
20 

\ 

'^— -- 

,'' 

S 

"-s 

S, 

PHASE 

^ 

\ 

\ 

■\ 

PHASE 
. CROSSOVER 

0 
-20 
-40 

GAIN-" 
CROSSOVER 

\ 

\ 

—  ?n  HR 

95=- 

360° 

■80 


-120 


160 


■200 


■240 


•280 


UJ 

_J 

z 
< 

LU 
lO 
-320  < 
I 
Q. 


-360 


-400 


■440 


10 


2  S      .-      2  5       .^3      2  5      ,^^      2  ^       105     2  ^       ,0«      '  '       10^ 


lO'^ 


w 


FREQUENCY    IN    CYCLES    PER   SECOND 

Fig.  14  — -  Loop  current  transmission  of  the  integrator. 

The  negative  feedback  in  the  integrator  has  been  shaped  by  means  of 
local  feedback  and  interstage  networks  as  described  in  Section  2.2.  The 
loop  current  transmission  has  been  calculated  from  (13),  (14),  (15),  and 
(A6)  and  is  plotted  in  Fig.  14.  The  transmission  is  determined  under  the 
assumption  that  the  alphas  of  the  transistors  are  0.985  and  the  alpha- 
cutoff  frequencies  are  three  megacycles.  Since  the  feedback  above  800 
cycles  per  second  falls  off  at  a  rate  of  9  db  per  octave,  the  analysis  in 
Section  4.1  using  (23),  is  conservative.  The  integrator  has  a  44°  phase 
margin  and  a  20  db  gain  margin.  In  order  to  insure  sufficient  feedback 
between  30  and  800  cycles  per  second  and  adequate  margins  against 
instability,  the  transistors  used  in  the  integrator  should  have  alphas  in 
the  range  0.98  to  0.99  and  alpha-cutoff  frequencies  equal  to  or  greater 
than  2.5  megacycles. 

The  silicon  diodes  Di  and  D2  are  rec^uired  in  order  to  prevent  the 
integrator  from  overloading.  For  output  voltages  between  —4.0  and  21 
volts  the  diodes  are  reverse  biased  and  represent  very  high  resistances,  of 
the  order  of  10,000  megohms.  If  the  output  voltage  does  not  lie  in  this 
range,  then  one  of  the  diodes  is  forward  biased  and  has  a  low  resistance, 
of  the  order  of  100  ohms.  The  integrator  is  then  effectively  a  dc  amplifier 
with  a  voltage  gain  of  approximately  0.1.  The  silicon  diodes  affect  the 


318 


THE   BELL   SYSTEM    TECHNICAL   JOURNAL,    MARCH    1956 


linearity  of  the  voltage  ramp  slightly  due  to  their  finite  reverse  resistances 
and  variable  shunt  capacities.  If  the  diodes  have  reverse  resistances 
greater  than  1000  megohms,  and  if  the  maximum  shunt  capacity  of  each 
diode  is  less  than  10  micromicrofarads  (capacity  with  minimum  reverse 
voltage),  then  the  diodes  introduce  negligible  error. 

As  stated  earlier,  the  integrator  generates  a  voltage  ramp  in  response 
to  a  voltage  step.  This  step  is  applied  through  a  transistor  switch  which 
is  actuated  by  a  square  wave  generator  capable  of  driving  the  transistor 
well  into  current  saturation.  Such  a  switch  is  required  because  the 
equivalent  generator  impedance  of  the  applied  step  voltage  must  be  very 
small.  A  suitable  circuit  arrangement  is  shown  in  Fig.  15.  For  the  par- 
ticular application  under  discussion  the  switch  *S  is  closed  for  5,000 
microseconds.  During  this  time,  the  voltage  E  =  —217  appears  at  the 
input  of  the  integrator.  At  the  end  of  this  time  interval,  the  transistor 
switch  is  opened  and  a  reverse  current  is  applied  to  the  feedback  con- 
denser C,  returning  the  output  voltage  to  —4.0  volts  in  about  2500  micro- 
seconds. An  alternate  way  of  specifying  a  low  impedance  switch  is  to  say 
that  the  voltage  across  it  be  close  to  zero.  For  the  transistor  switch,  con- 
nected as  shown  in  Fig.  15,  this  means  that  its  collector  voltage  be  within 


FIRST  STAGE 

OF   DC 

AMPLIFIER 


10.5V 


50  K  150 K 

' — WV-HVW 


RESIDUAL 
VOLTAGE    BALANCE 


(TO  AZS) 


Fig.  15  —  Input  circuit  arrangement  of  the  integrator. 


TRANSISTOR   CIRCUITS    FOR   ANALOG    AND    DIGITAL   SYSTEMS         319 

one  millivolt  of  ground  potential  during  the  time  the  transistor  is  in 
saturation.  Xow,  it  has  been  shown  that  when  a  junction  transistor  in 
the  common  emitter  connection  is  driven  into  current  saturation,  the 
minimum  voltage  between  collector  and  emitter  is  theoretically  equal  to 

—  in  -  (25) 

q         oci 

where  k  is  the  Boltzmann  constant,  T  is  the  absolute  temperature,  q  is 
the  charge  of  an  electron  ((kT/q)  =  26  millivolts  at  room  temperature), 
and  ai  is  the  inverse  alpha  of  the  transistor,  i.e.,  the  alpha  with  the 
emitter  and  collector  interchanged.  There  is  an  additional  voltage  drop 
across  the  transistor  due  to  the  bulk  resistance  of  the  collector  and 
emitter  regions  (including  the  ohmic  contacts).  A  symmetrical  alloy 
junction  transistor  with  an  alpha  close  to  unity  is  an  excellent  switch 
because  both  the  collector  to  emitter  voltage  and  the  collector  and  emit- 
ter resistances  are  very  small. 

At  the  present  time,  a  reasonable  value  for  the  residual  voltage*  be- 
tween the  collector  and  emitter  is  5  to  10  millivolts.  This  voltage  can  be 
eliminated  by  returning  the  emitter  of  the  transistor  switch  to  a  small 
negative  potential.  This  method  of  balancing  is  practical  because  the 
voltage  between  the  collector  and  emitter  of  the  transistor  does  not 
change  by  more  than  1.0  millivolt  over  a  temperature  range  of  0°C  to 
50°C. 

4.3.  Automatic  Zero  Set  of  the  Integrator 

A  serious  problem  associated  with  the  transistor  integrator  is  drift. 
The  drift  is  introduced  by  two  sources;  variations  in  the  base  current  of 
the  first  transistor  stage  and  variations  in  the  base  to  emitter  potential 
of  the  first  stage  wdth  temperature.  In  order  to  reduce  the  drift,  the 
input  resistor  R  and  the  feedback  condenser  C  must  be  dissociated  from 
the  base  current  and  base  to  emitter  potential  of  the  first  transistor  stage. 
This  is  accomplished  by  placing  a  blocking  condenser  Cb  between  point 
T  and  the  base  of  the  first  transistor  as  shown  in  Fig.  15.  An  automatic 
zero  set  circuit  is  required  to  maintain  the  voltage  at  point  T  equal  to 
zero  volts.  This  AZS  circuit  uses  a  magnetic  modulator  known  as  a 
"magnettor."^^ 

A  block  diagram  of  the  AZS  circuit  is  shown  in  Fig.  16.  The  dc  drift 
current  at  the  input  of  the  amplifier  is  applied  to  the  magnettor.  The 
carrier  current  required  by  the  magnettor  is  supplied  by  a  local  transistor 

*  The  inverse  alphas  of  the  transistors  used  in  this  application  were  greater 
than  0.95. 


320 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    1956 


oscillator.  The  useful  output  of  the  magnettor  is  the  second  harmonic  of 
the  carrier  frequency.  The  amplitude  of  the  second  harmonic  signal  is 
proportional  to  the  magnitude  of  the  dc  input  current  and  the  phase  of 
the  second  harmonic  signal  is  determined  by  the  polarity  of  the  dc  input 
current.  The  output  voltage  of  the  magnettor  is  applied  to  an  active 
filter  which  is  tuned  to  the  second  harmonic  frequency.  The  signal  is 
then  amplified  in  a  tuned  amplifier  and  applied  to  a  diode  gating  circuit. 
Depending  on  the  polarity  of  the  dc  input  current,  the  gating  circuit 
passes  either  the  positive  or  negative  half  cycle  of  the  second  harmonic 
signal.  In  order  to  accomplish  this,  a  square  wave  at  a  repetition  rate 
equal  to  that  of  the  second  harmonic  signal  is  derived  from  the  carrier 
oscillator  and  actuates  the  gating  circuit. 

A  circuit  diagram  of  the  AZS  circuit  is  shown  in  Figs.  17(a)  and  17(b). 
The  various  sections  of  the  circuit  are  identified  with  the  blocks  shown 
in  Fig.  16.  The  active  filter  is  adjusted  for  a  Q  of  about  300,  and  the  gain 
of  the  active  filter  and  tuned  amplifier  is  approximately  1000.  The  AZS 
circuit  provides  ±1.0  volt  of  dc  output  voltage  for  ±0.05  microamperes 
of  dc  input  current.  The  maximum  sensitivity  of  the  circuit  is  limited 
to  ±0.005  microamperes  because  of  residual  second  harmonic  generation 
in  the  magnettor  with  zero  input  current. 

When  the  transistor  integrator  is  used  together  with  the  magnettor 
AZS  circuit,  the  slope  of  the  voltage  ramp  is  maintained  constant  to 
within  one  part  in  8,000  over  a  temperature  range  of  20°C  to  40°C. 

5.0.  The  Voltage  Comparator 

The  voltage  comparator  is  one  of  the  most  important  circuits  used  in 
analog  to  digital  converters.  The  comparator  indicates  the  exact  time 
that  an  input  waveform  passes  through  a  predetermined  reference  level. 
It  has  been  common  practice  to  use  a  vacuum  tube  blocking  oscillator 
as  a  voltage  comparator. ^^  Due  to  variations  in  the  contact  potential, 
heater  voltage,  and  transconductance  of  the  vacuum  tube,  the  maximum 


DC 
INPUT 


AC 

MAGNETTOR 

ACTIVE 
FILTER 

\ 

GATING 
CIRCUIT 

^ 

A 

■~ 

OSCILLATOR 

GATING 
PULSE 

DC 

OUTPUT 


Fig.  16  —  Block  diagram  of  AZS  circuit. 


TRANSISTOR    CIRCUITS    FOR   ANALOG    AND    DIGITAL   SYSTEMS         321 

accuracy  of  the  circuit  is  limited  to  about  ±100  millivolts.  By  taking 
advantage  of  the  properties  of  semiconductor  devices,  the  transistor 
blocking  oscillator  comparator  can  be  designed  to  have  an  accuracy  of 
±5  millivolts  throughout  a  temperature  range  of  20°C  to  40°C. 

5.1.  General  Descri'ption  of  the  Voltage  Comparator 

Fig.  18  shows  a  simplified  circuit  diagram  of  the  voltage  comparator. 
Except  for  the  silicon  junction  diode  D\  ,  this  circuit  is  essentially  a 
transistor  blocking  oscillator.  For  the  purpose  of  analysis,  assume  that 
the  reference  voltage  Vee  is  set  equal  to  zero.  When  the  input  voltage  V, 
is  large  and  negative,  the  silicon  diode  Di  is  an  open  circuit  and  the  jiuic- 
tion  transistor  has  a  collector  current  determined  by  Rb  and  Ebb  [Expres- 
sion (18)].  The  base  of  the  transistor  resides  at  approximately  —0.2 
volts.  As  the  input  voltage  Vi  approaches  zero,  the  reverse  bias  across 
the  diode  Di  decreases.  At  a  critical  value  of  Vi  (a  small  positive  poten- 
tial), the  dynamic  resistance  of  the  diode  is  small  enough  to  permit  the 
circuit  to  become  unstable.  The  positive  feedback  provided  by  trans- 
1  former  Ti  forces  the  transistor  to  turn  off  rapidly,  generating  a  sharp 
I  output  pulse  across  the  secondary  of  transformer  T-z  .  When  Vi  is  large 
and  positive,  the  diode  Di  is  a  low  impedance  and  the  transistor  is  main- 
tained cutoff.  In  order  to  prevent  the  comparator  from  generating  more 
than  one  output  pulse  during  the  time  that  the  circuit  is  unstable,  the 
natural  period  of  the  circuit  as  a  blocking  oscillator  must  be  properly 
chosen.  Depending  on  this  period,  the  input  voltage  waveform  must 
have  a  certain  minimum  slope  when  passing  through  the  reference  level 
in  order  to  prevent  the  circuit  from  misfiring. 

I      The  comparator  has  a  high  input  impedance  except  during  the  switch- 
1  ing  interval.*  When  Vi  is  negative  with  respect  to  the  reference  level,  the 
\  input  impedance  is  equal  to  the  impedance  of  the  reverse  biased  silicon 
i  diode.  When  Vi  is  positive  with  respect  to  the  reference  level,  the  input 
I  impedance  is  equal  to  the  impedance  of  the  reverse  biased  emitter  and 
!  collector  junctions  in  parallel.   This   impedance  is  large   if   an   alloy 
;  junction  transistor  is  used.  During  the  switching  interval  the  input  im- 
■  pedance  is  equal  to  the  impedance  of  a  forward  biased  silicon  diode  in 
series  with  the  input  impedance  of  a  common  emitter  stage  (approxi- 
mately 1,000  ohms).  This  loading  effect  is  not  too  serious  since  for  the 
circuit  described,  the  switching  interval  is  less  than  0.5  microseconds. 

The  voltage  comparator  shown  in  Fig.  18  operates  accurately  on 
voltage  waveforms  with  positive  slopes.  The  voltage  comparator  will 
operate  accurately  on  waveforms  with  negative  slopes  if  the  diode  and 

*  The  switching  interval  is  the  time  required  for  the  transistor  to  turn  off. 


322  THE    BELL    SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


note:  all  capacitors  and  inductors 
IN  tuned  circuits  have  a 

tolerance    of    ±0.1% 


Fig.  17(a)  —  AZS  circuit. 

battery  potentials  are  reversed  and  if  an  n-p-n  junction  transistor  is 
used. 


5.2.  Factors  Determining  the  Accuracy  of  the  Voltage  Comparator 

Fig.  19  shows  the  ac  equivalent  circuit  of  the  voltage  comparator.  In 
the  equivalent  circuit  Ri  is  the  dynamic  resistance  of  the  diode  Di ,  Rg 
is  the  source  resistance  of  the  input  voltage,  and  R2  is  the  impedance  of 


TRANSISTOR   CIRCUITS   FOR  ANALOG   AND   DIGITAL   SYSTEMS 


323 


the  load  R^  as  it  appears  at  the  primary  of  the  transformer  T2 .  Ri  is  a 
function  of  the  dc  voltage  across  the  diode  Z)i .  At  a  prescribed  value  of 
Ri ,  the  comparator  circuit  becomes  unstable  and  switches.  The  relation- 
ship between  this  critical  value  of  Ri  and  the  transistor  and  circuit 
parameters  is  obtained  by  evaluating  the  characteristic  equation  for  the 
circuit  and  by  determining  the  relationship  which  the  coefficients  of  the 
equation  must  satisfy  in  order  to  have  a  root  of  the  equation  lie  in  the 
right  hand  half  of  the  complex  frequency  plane.  To  a  good  approxima- 
tion, the  critical  value  of  Ri  is  given  by  the  expression 


R,  -\-R„  +  n  = 


Mao 


RiCc  -\- 


(26) 


N'^Rr 


where  M  is  the  mutual  inductance  of  transformer  Ti  and  R2  —  ly  h^l 
Since  the  transistor  parameters  which  appear  in  expression  (26)  have  only 
a  small  variation  with  temperature,  the  critical  value  of  Ri  is  independent 
of  temperature  (to  a  first  approximation). 

It  will  now  be  shown  that  the  comparator  can  be  designed  for  an  ac- 
curacy of  ±5  millivolts  throughout  a  temperature  range  of  20°C  to  40°C. 
In  order  to  establish  this  accuracy  it  will  be  assumed  that  the  critical 
value  of  7^1  is  equal  to  30,000  ohms.  This  assumption  is  based  on  the 


30/iF 


TO    LC    FILTER 

IN     MAGNETTOR 

NPUT    CIRCUIT 


4/iF 


+33V 


I+33V 

Fig.  17(b),  900-cycle  carrier  oscillator. 


324  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    1956 


data  displayed  in  Fig.  20  which  gives  the  volt-ampere  characteristics  of  a 
silicon  diode  measured  at  20°C  and  40°C.  Throughout  this  temperature 
range,  the  diode  voltage  corresponding  to  the  critical  resistance  of 
30,000  ohms  changes  by  about  30  millivolts.  Fortunately,  part  of  this 
voltage  variation  with  temperature  is  compensated  for  by  the  variation 
in  voltage  Vb-e  between  the  base  and  emitter  of  the  junction  transistor. 
From  Fig.  18, 


V,    =    Vo    -    Vb-e  +    Ve 


(27) 


For  perfect  compensation  (Vi  independent  of  temperature),  Vb-e  should 
have  the  same  temperature  variation  as  the  diode  voltage  Vd  .  Experi- 


REFERENCE 
I  LEVEL 

-I       ADJUSTMENT       i+ 


Fig.  18  —  Simplified  circuit  diagram  of  voltage  comparator. 


Fig.  19  —  Equivalent  circuit  of  voltage  comparator. 


TRANSISTOR   CIRCUITS   FOR   ANALOG   AND   DIGITAL   SYSTEMS 


325 


0.7 


_) 
O 
> 


0.6 


R,  =  30,000  OHMS 


20°C 


> 


u:o.5 

< 

I- 
_l 

o 
> 

o 


0.3 


2  3  4  5  6 

DIODE   CURRENT,  Ip,  IN    MICROAMPERES 

Fig.  20  —  Volt-ampere  characteristic  of  a  silicon  junction  diode. 

mentally  it  is  found  that  Yh-e  for  germanium  junction  transistors  varies 
by  about  20  millivolts  throughout  the  temperature  range  of  20°C  to 
40°C.  Consequently,  the  variation  in  Yi  at  which  the  circuit  switches  is 
±5  millivolts. 

It  is  apparent  from  Fig.  20  that  the  accuracy  of  the  comparator  in- 
creases slightly  for  critical  values  of  R\  greater  than  30,000  ohms,  but 
decreases  for  smaller  values.  For  example,  the  accuracy  of  the  comparator 
is  ±10  millivolts  for  a  critical  value  of  U\  equal  to  5,000  ohms.  In  gen- 
eral, the  critical  value  of  R\  should  be  chosen  between  5,000  and  100,000 
ohms. 


5.3.  A  Practical  Yoltage  Comparator 

Fig.  21  shows  the  complete  circuit  diagram  of  a  voltage  comparator. 
The  circuit  is  designed  to  generate  a  sharp  output  pulse*  when  the  input 
voltage  waveform  passes  through  the  reference  level  (set  by  Yee)  with  a 
positive  slope.  The  pulse  is  generated  by  the  transistor  switching  from 
the  "on"  state  to  the  "off"  state.  To  a  first  approximation  the  amplitude 
of  the  output  pulse  is  proportional  to  the  transistor  collector  current 
during  the  "on"  state.  When  the  input  voltage  waveform  passes  through 
the  reference  level  with  a  negative  slope  an  undesirable  negative  pulse  is 
generated.  This  pulse  is  eliminated  by  the  point  contact  diode  D2 . 

The  voltage  comparator  is  an  unstable  circuit  and  has  the  properties 

*  For  the  circuit  values  shown  in  Fig.  21,  the  output  pulse  has  a  peak  amplitude 
of  about  6  volts,  a  rise  time  of  0.5  microseconds,  and  a  pulse  width  of  about  2.0 
microseconds. 


32G 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


of  a  free  running  blocking  oscillator  after  the  input  voltage  Vi  passes 
through  the  reference  level.  After  a  period  of  time  the  transistor  will 
return  to  the  "on"  state  unless  the  voltage  Vi  is  sufficiently  large  at  this 
time  to  prevent  switching.  In  order  to  minimize  the  required  slope  of  the 
hiput  waveform  the  time  interval  between  the  instant  Vi  passes  through 
the  reference  level  and  the  instant  the  transistor  would  naturally  switch 
to  the  "on"  state  must  be  maximized.  This  time  intei-val  can  be  con- 
trolled by  connecting  a  diode  D3  across  the  secondary  winding  of  trans- 
former Ti  .  When  the  transistor  turns  off,  the  current  which  was  flowing 
through  the  secondary  of  transformer  Ti(Ic)  continues  to  flow  through 
the  diode  D3  so  that  L2  and  D3  form  an  inductive  discharge  circuit.  The 
point  contact  diode  D3  has  a  forward  dynamic  resistance  of  less  than  10 
ohms  and  a  forward  voltage  drop  of  0.3  volt.  If  the  small  forward  re- 
sistance of  the  diode  is  neglected,  the  time  required  for  the  current  in  the 
circuit  to  fall  to  zero  is 


T  = 


0.3 


(28) 


During  the  inductive  transient,  0.3  volt  is  induced  into  the  primary  of 
transformer  Ti  (since  N  =  1)  maintaining  the  transistor  cutoff.  The 
duration  of  the  inductive  transient  can  be  made  as  long  as  desired  by 
increasing  L2 .  However,  there  is  the  practical  limitation  that  increasing 
L2  also  increases  the  leakage  inductance  of  transformer  Ti ,  and  in  turn, 


I 


I 


-4.5V 


5.1K 


250A 


:iD2 


>3K 


A-l- 


OUTPUT 
PULSE 


V- 


INPUT 
WAVEFORM 


PULSE 
AMPLITUDE^, 
ADJUSTMENT^ 


•^ 


2.5  MEG  POT. 
I- 


jr 


ee' 


Ij,  =  4   MILS 

L,    =  L2=  5  MILLIHENRIES 

L',    =  L2=  5  MILLIHENRIES 

COEFFICIENT    OF 

COUPLING  =  0.99 


REFERENCE 
g^    LEVEL 

''adjustment 

MA 1 


I 
I 

-46V 


I 
I 

-t-1.5V 


100  OHM 
POT. 


I 
I 

-1.5V 


Fig.  21  —  Voltage  comparator. 


TRANSISTOR    CIRCUITS    FOR   ANALOG    AND    DIGITAL   SYSTEMS 


327 


increases  the  switching  time.  The  circuit  shown  in  Figure  21  does  not 
misfire  when  used  with  voltage  waveforms  having  slopes  as  small  as  25 
millivolts  per  microsecond,  at  the  reference  level. 


6.0.  A  TRANSISTOR  VOLTAGE  ENCODER 


6.1.  Circuit  Arrangement 


The  transistor  circuits  previously  described  can  be  assembled  into  a 
voltage  encoder  for  translating  analog  voltages  into  equivalent  time 
intervals.  This  encoder  is  especially  useful  for  converting  analog  informa- 
,  tion  (in  the  form  of  a  dc  potential)  into  the  digital  code  for  processing 
in  a  digital  system.  Fig.  22  shows  a  simplified  block  diagram  of  the 
encoder.  The  voltage  I'amp  generated  by  the  integrator  is  applied  to 
amplitude  selector  number  one  and  to  one  input  of  a  summing  amplifier. 
The  amplitude  selector  is  a  dc  amplifier  which  amplifies  the  voltage  ramp 
in  the  vicinity  of  zero  volts.  Voltage  comparator  number  one,  which 
follows  the  amplitude  selector,  generates  a  sharp  output  pulse  at  the 
exact  instant  of  time  that  the  voltage  ramp  passes  through  zero  volts. 

The  analog  input  voltage,  which  has  a  value  between  0  and  —15 
volts,*  is  applied  to  the  second  input  of  the  summing  amplifier.  The 
output  voltage  of  the  summing  amplifier  is  zero  whenever  the  ramp 


INTEGRATOR 


N0.1 


N0.1 


3000^65 


SUMMING 
AMPLIFIER 


AMPLITUDE 
SELECTORS 


VOLTAGE 
COMPARATORS 


ANALOG 

INPUT   VOLTAGE 

0-^-16V 


N0.2 


N0.2 


Fig.  22  — •  Simplified  block  diagram  of  voltage  encoder. 


*  If  the  analog  input  voltage  does  not  lie  in  this  range,  then  the  voltage  gain 
of  the  summing  amplifier  must  be  set  so  that  the  analog  voltage  at  the  output  of 
the  summing  amplifier  lies  in  the  voltage  range  between  0  and  +15  volts. 


328 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


voltage  is  equal  to  the  negative  of  the  input  analog  voltage.  At  this 
instant  of  time  the  second  voltage  comparator  generates  a  sharp  output 
pulse.  The  time  interval  between  the  two  output  pulses  is  proportional 
to  the  analog  input  voltage  if  the  voltage  ramp  is  linear  and  has  a  con- 
stant slope  at  all  times. 

6.2.  The  Amplitude  Selector  i 

The  amplitude  selector  increases  the  slope  of  the  input  voltage  wave- 
form (in  the  vicinity  of  zero  volts)  sufficiently  for  proper  operation  of  the 
voltage  comparator.  The  amplitude  selector  consists  of  a  limiter  and  a 
dc  feedback  amplifier  as  shown  in  Fig.  23.  The  two  oppositely  poled 
silicon  diodes  Di  and  D2 ,  limit  the  input  voltage  of  the  dc  amplifier  to 
about  ±0.65  volts.  The  dc  amplifier  has  a  voltage  gain  of  thirty,  and  so 
the  maximum  output  voltage  of  the  amplitude  selector  is  limited  to 
about  ±19.5  volts.  The  net  voltage  gain  between  the  input  and  output 
of  the  amplitude  selector  is  ten. 

The  principal  requirement  placed  on  the  dc  amplifier  is  that  the  input 
current  and  the  output  voltage  be  zero  when  the  input  voltage  is  zero. 
This  is  accomplished  by  placing  a  blocking  condenser  Cb  between  point 
T  and  the  base  of  the  first  transistor  stage,  and  by  using  an  AZS  circuit 
to  maintain  point  T  at  zero  volts.  The  dc  and  AZS  amplifiers  are  identical 
in  configuration  to  the  amplifiers  shown  in  Fig.  12.  The  dc  amplifier  is 


50  K 

-VvV 


50  K 


:|N 


D 


1:: 


SILICON 
DIODES 


Dp 


1.5  MEG 


Cb 

250 /ZF 


500  K 


I 


OUT 


I V^^ »— AAA^ 

50  K  1.5  MEG 


-1 


Fig.  23  —  Block  diagram  of  the  amplitude  selector. 


TRANSISTOR   CIRCUITS    FOR   ANALOG   AND    DIGITAL   SYSTEMS         329 

designed  to  have  about  15.6  db  less  feedback  than  that  shown  in  Fig.  10 
since  this  amount  is  adequate  for  the  present  purpose. 

The  bandwidth  of  the  dc  ampHfier  is  only  of  secondary  importance 
because  the  phase  shifts  introduced  by  the  two  amplitude  selectors  in 
the  voltage  encoder  tend  to  compensate  each  other. 

6.3.  Experimental  Results 

The  accuracy  of  the  voltage  encoder  is  determined  by  applying  a 
precisely  measured  voltage  to  the  input  of  the  summing  amplifier  and  by 
measuring  the  time  interval  between  the  two  output  pulses.  The  maxi- 
mum error  due  to  nonlinearities  in  the  summing  amplifier  and  the  voltage 
ramp  is  less  than  ±0.5  microseconds  for  a  maximum  encoding  time  of 
3,000  microseconds.  An  additional  error  is  introduced  by  the  noise  voltage 
generated  in  the  first  transistor  stage  of  the  summing  amplifier.  The 

!  RMS  noise  voltage  at  the  output  of  the  summing  amplifier  is  less  than 
0.5  millivolts.  This  noise  voltage  produces  an  RMS  jitter  of  0.25  micro- 

I  seconds  in  the  position  of  the  second  voltage  comparator  output  pulse. 

;  The  over-all  accuracy  of  the  voltage  encoder  is  one  part  in  4,000  through- 

'  out  a  temperature  range  of  20°C  to  40°C. 

1 
I 

i  ACKNOWLEDGEMENTS 

! 

I  The  author  wishes  to  express  his  appreciation  to  T.  R.  Finch  for  the 
^  advice  and  encouragement  received  in  the  course  of  this  work.  D.  W. 
!  Grant  and  W.  B.  Harris  designed  and  constructed  the  magnettor  used 
'  in  the  AZS  circuit  of  the  integrator. 

I  Appendix  I 

I  RELATIONSHIP  BETWEEN  RETURN  DIFFERENCE  AND  LOOP  CURRENT 
i       TRANSMISSION 

}  In  order  to  place  the  stability  analysis  of  the  transistor  feedback  ampli- 
fier on  a  sound  basis,  it  is  desirable  to  use  the  concept  of  return  differ- 
ence. It  will  be  shown  that  a  measurable  quantity,  called  the  loop  current 
transmission,  can  be  related  to  the  return  difference  of  aZc  with  reference 
Ve  .*•  t  In  Fig.  24,  N  represents  the  complete  transistor  network  exclusive 
of  the  transistor  under  consideration.  The  feedback  loop  is  broken  at 
the  input  to  the  transistor  by  connecting  all  of  the  feedback  paths  to 

*  In  this  appendix  it  is  assumed  that  the  transistor  under  consideration  is  in 
the  common  emitter  connection.  The  discussion  can  be  readily  extended  to  the 
other  transistor  connections. 

t  This  fact  was  pointed  out  by  F.  H.  Tendick,  Jr. 


330 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


Te+rb 


^6^4 (    'V 


-aZcLb 


N 

COMPLETE 
AMPLIFIER 
EXCLUSIVE 

OF    THE 
TRANSISTOR 
IN   QUESTION 


Fig.  24  —  Measurement  of  loop  current  transmission. 


ground  through  a  resistance  (/•<;  +  n)  and  a  voltage  r  J4  •  Using  the 
nomenclature  given  in  Reference  8,  the  input  of  the  complete  circuit  is 
designated  as  the  first  mesh  and  the  output  of  the  complete  circuit  is 
designated  as  the  second  mesh.  The  input  and  output  meshes  of  the 
transistor  under  consideration  are  designated  3  and  4,  respectively.  The 
loop  current  transmission  is  equal  to  I3',  the  total  returned  current  when 
a  unit  input  current  is  applied  to  the  base  of  the  transistor. 

The  return  difference  for  reference  Ve  is  equal  to  the  algebraic  differ- 
ence* between  the  unit  input  current  and  the  returned  current  h'.  1 3  is 
evaluated  by  multiplying  the  open  circuit  voltage  in  mesh  4  (produced 
by  the  unit  base  current)  by  the  backward  transmission  from  mesh  4  to 
mesh  3  with  zero  forward  transmission  through  the  transistor  under 
consideration.  The  open  circuit  voltage  in  mesh  4  is  equal  to  (re  —  aZc). 
The  backward  transmission  is  determined  with  the  element  aZc ,  in  the 
fourth  row,  third  column  of  the  circuit  determinant,  set  equal  to  Ve . 
Hence,  the  return  difference  is  expressed  as 

A43 


Fr'    =  1  +  {aZc  -  re) 


(Al)t 


Fr'      = 


A''*  +  {aZc  -  r.)A 


43 


ir', 


(A2) 


Fr'.= 


A^'' 


=    1+    Tr' 


(A3) 


The  relative  return  ratio  Tr',  is  equal  to  the  negative  of  the  loop  current 
transmission  and  can  be  measured  as  shown  in  Fig.  24.  The  voltage  reh 
takes  into  account  the  fact  that  the  junction  transistor  is  not  perfectly 


*  The  positive  direction  for  the  returned  current  is  chosen  so  that  if  the  original 
circuit  is  restored,  the  returned  current  flows  in  the  same  direction  as  the  input 
current. 

t  A''«  is  the  network  determinant  with  the  element  aZc  in  the  fourth  row,  third 
column  of  the  circuit  determinant  set  equal  to  r,  . 


TRANSISTOR    CIRCUITS    FOR    ANALOG    AND    DIGITAL   SYSTEMS         331 


unilateral.  Fortunately,  in  many  applications,  this  voltage  can  be  neg- 
lected even  at  the  gain  and  phase  crossover  frequencies. 

In  the  case  of  single  loop  feedback  amplifiers.  A""*  will  not  have  any 
zeros  in  the  right  hand  half  of  the  complex  frequency  plane.  A  study  of 
the  stability  of  the  amplifier  can  then  be  based  on  F^-,  or  T^-,  . 

Appendix  II 

INTERSTAGE   NETWORK   SHAPING 

This  appendix  presents  the  analysis  of  the  circuit  shown  in  Fig.  7(a). 
The  input  impedance  of  the  common  emitter  connected  junction  tran- 
sistor is  given  by  the  expression 

^iNPUT  =  n-\-  re(l  -  Gl)  (A4) 

where  Gi  is  the  current  transmission  of  the  common  emitter  stage,  ex- 
pression (13).  The  current  transmission  A  of  the  complete  circuit  is  equal 
to 

A  =  ^  =  ^ 

I\  Zz   -\-    ^  IN  PUT 


G, 


(A5) 


where  Z3  =  i?3  +  V^  +  (l/p<^3).  Combining  (13),  (A4),  and  (A5)  yields 


ao 


A  = 


1  —  ao  +  5 


1  + 


C03 


+  V 


\  W5/    I,  Wl 


(A6) 


+  p^ 


W5         ,     CsOO^in   +  Te   -\-   R3) 


_C0iC03- 


+ 


PCO5 


where 


WaWc(l    —    tto    -}-    6)    J      '      CO3^C0aC0c(l     —    ^Q    "j-    6)  j 
^    ^    Rl    +  Te 

COl    = 
Wc    = 


CO3 


OJs    = 


(1  -  ao  +  5) 

1  +  6  _^    1 

1 

(R^  +  r,)Co 
1 

1 

~.    .             ^« 

C 

(1  -  ao 

+  5)J^ 

332  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    1956 

Expression  (A6)  is  valid  if  l/ws  ^  1/coi  +  RzCz .  The  denominator  of  the 
expression  indicates  a  falUng  6  db  per  octave  asymptote  with  a  corner, 
frequency  at  ws  .  The  second  factor  in  the  denominator  can  be  approxi- 
mated bj^  a  falHng  6  db  per  octave  asymptote  with  a  corner  frequency  at 


COl 


1  ^^ 

n  + 


(1  -  ao  +  5) 


] 


n   -\-    Te    -^    Rz   -\-    W1L3 

pkis  additional  phase  and  amplitude  contributions  at  higher  f recjuencies 
due  to  the  y  and  p  terms.  If 


COzCzRz 


then  the  circuit  has  a  rising  12  db  per  octave  asymptote  with  a  corner 
frequency  at  C03  .  Fig.  7(b)  shows  the  amplitude  and  phase  of  the  current 
transmission. 


REFERENCES 

1.  Felker,  J.  H.,  Regenerative  Amplifier  for  Digital  Computer  Applications, 

Proc.  I.R.E.,  pp.  1584-1596,  Nov.,  1952. 

2.  Korn,  G.  A.,  and  Korn,  T.  M.,  Electronic  Analog  Computers,  McGraw-Hil 

Book  Company,  pp.  9-19. 

3.  Wallace,  R.  L.  and  Pietenpol,  W.  J.,  Some  Circuit  Properties  and  Applications 

of  n-p-n  Transistors,  B.  S.T.J. ,  30,  pp.  530-563,  July,  1951. 

4.  Shockley,  W.,  Sparks,  M.  and  Teal,  G.  K.,  The  p-n  Junction  Transistor, 

Physical  Review,  83,  pp.  151-162,  July,  1951. 

5.  Pritchard,  R.  L.,  Frequenc}'  Variation  of  Current-Amplification  for  Junction 

Transistors,  Proc.  I.R.E.,  pp.  1476-1481,  Nov.,  1952. 

6.  Early,  J.  M.,  Design  Theory  of  Junction  Transistors,  B.S.T.J.,  32,  pp.  1271- 

1312,  Nov.,  1953. 

7.  Sziklai,  G.  C,  Symmetrical  Properties  of  Transistors  and  Their  Applications, 

Proc.  I.R.E.,  pp.  717-724,  June,  1953. 

8.  Bode,  H.  W.,  Network  Analysis  and  Feedback  Amplifier  Design,  Van  Nos- 

trand  Co.,  Inc.,  Chapter  IV. 

9.  Bode,  H.  W.,  Op.  Cit.,  pp.  66-69. 

10.  Bode,  H.  W.,  Op.  Cit.,  pp.  162-164. 

11.  Bargellini,  P.  M.  and  Herscher,  M.  B.,  Investigation  of  Noise  in  Audio  Fre- 

quency Amplifiers  Using  Junction  Transistors,  Proc.  I.R.E.,  pp.  217-226,' 
Feb.,  1955. 

12.  Bode,  H.  W.,  Op.  Cit.,  pp.  464-468,  and  pp.  471-473. 

13.  Keonjian,  E.,  Temperature  Compensated  DC  Transistor  Amplifier,  Proc: 

I.R.E.,  pp.  661-671,  April,  1954. 

14.  Kretzmer,  E.  R.,  An  Amplitude  Stabilized  Transistor  Oscillator,  Proc.  I.R.E.,« 

pp.  391-401,  Feb.,  1954.  i 

15.  Goldberg,  E.  A.,  Stabilization  of  Wide-Band  Direct-Current  Amplifiers  for 

Zero  and  Gain,  R.C.A.  Review,  June,  1950. 

16.  Ebers,  J.  J.  and  Moll,  J.  L.,  Large  Signal  Behavior  of  Junction  Transistors. 

Proc.  I.R.E.,  pp.  1761-1772,  Dec,  1954. 

17.  Manlej',  J.  M.,  Some  General  Properties  of  Magnetic  Amplifiers,  Proc.  I.R.K. 

March,  1951. 

18.  M.I.T.,  Waveforms,  Volume  19  of  the  Radiation  Laboratories  Series.  McGraw 

Hill  Book  Company,  pp.  342-344. 


Electrolytic  Shaping  of  Germanium 
,  and  Silicon 

^  By  A.  UHLIR,  JR. 

i  (Manuscript  received  November  9,  1955) 

Properties  of  electrolyte-semiconductor  barriers  are  described,  with  em- 
phasis on  germanium.  The  use  of  these  barriers  in  localizing  electrolytic 
!  etching  is  discussed.  Other  localization  techniques  are  mentioned.  Electro- 
lytes for  etching  germanium  and  silicon  are  given. 

I 

INTRODUCTION 

I 

I      Mechanical  shaping  techniques,  such  as  abrasive  cutting,  leave  the 
surface  of  a  semiconductor  in  a  damaged  condition  which  adversely 
affects  the  electrical  properties  of  p-n  junctions  in  or  near  the  damaged 
j  material.  Such  damaged  material  may  be  removed  by  electrolytic  etch- 
ing. Alternatively,  all  of  the  shaping  may  be  done  electrolytically,  so 
that  no  damaged  material  is  produced.  Electrolytic  shaping  is  particu- 
[  larly  well  suited  to  making  devices  with  small  dimensions. 
I     A  discussion  of  electrolytic  etching  can  conveniently  be  divided  into 
[■  two  topics  —  the  choice  of  electrolyte  and  the  method  of  localizing  the 
ji  etching  action  to  produce  a  desired  shape.  It  is  usually  possible  to  find 
1  an  electrolyte  in  which  the  rate  at  which  material  is  removed  is  accurately 
proportional  to  the  current.  For  semiconductors,  just  as  for  metals,  the 
I  choice  of  electrolyte  is  a  specific  problem  for  each  material ;  satisfactory 
j  electrolytes  for  germanium  and  silicon  will  be  described. 

The  principles  of  localization  are  the  same,  whatever  the  electrolyte 

used.  Electrolytic  etching  takes  place  where  current  flows  from  the 

semiconductor  to  the  electrolyte.  Current  flow  may  be  concentrated  at 

I  certain  areas  of  the  semiconductor-electrolyte  interface  by  controlling 

the  flow  of  current  in  the  electrolyte  or  in  the  semiconductor. 

LOCALIZATION    IN    ELECTROLYTE 

Localization  techniques  involving  the  electrolytic  current  are  appli- 
cable to  both  metals  and  semiconductors.  In  some  of  these  techniques, 

333 


334  THE   BELL   SYSTEM   TECHNICAL  JOURNAL,    MARCH    1956 

the  localization  is  so  effective  that  the  barrier  effects  found  with  n-type 
semiconductors  can  be  ignored;  if  not,  the  barrier  can  be  overcome  by 
light  or  heat,  as  will  be  described  below. 

If  part  of  the  work  is  coated  with  an  insulating  varnish,  electrolytic 
etching  will  take  place  only  on  the  uncoated  surfaces.  This  technique, 
often  called  "masking,"  has  the  limitation  that  the  etching  undercuts 
the  masking  if  any  considerable  amount  of  material  is  removed.  The  i 
same  limitation  applies  to  photoengraving,  in  which  the  insulating  coat- 
ing is  formed  by  the  action  of  light. 

The  cathode  of  the  electrolytic  cell  may  be  limited  in  size  and  placed 
close  to  the  work  (which  is  the  anode).  Then  the  etching  rate  will  be 
greatest  at  parts  of  the  work  that  are  nearest  the  cathode.  Various 
shapes  can  be  produced  by  moving  the  cathode  with  respect  to  the  I 
work,  or  by  using  a  shaped  cathode.  For  example,  a  cathode  in  the  form  | 
of  a  wire  has  been  used  to  slice  germanium. 

Instead  of  a  true  metallic  cathode,  a  "virtual  cathode"  may  be  used 
to  localize  electrolysis.^  In  this  technique,  the  anode  and  true  cathode 
are  separated  from  each  other  by  a  nonconducting  partition,  except  for 
a  small  opening  in  the  partition.  As  far  as  localization  of  current  to  the 
anode  is  concerned,  the  small  opening  acts  like  a  cathode  of  equal  size 
and  so  is  called  a  virtual  cathode.  The  nonconducting  partition  may 
include  a  glass  tube  drawn  down  to  a  tip  as  small  as  one  micron  diameter 
but  nevertheless  open  to  the  flow  of  electrolytic  current.  With  such  a 
tip  as  a  virtual  cathode,  micromachining  can  be  conducted  on  a  scale 
comparable  to  the  wavelength  of  visible  light.  A  general  advantage  of 
the  virtual  cathode  technique  is  that  the  cathode  reaction  (usually 
hydrogen  evolution)  does  not  interfere  with  the  localizing  action  nor 
with  observation  of  the  process.  :| 

In  the  jet-etching  technique,  a  jet  of  electrolyte  impinges  on  the 
work.^'*  The  free  streamlines  that  bound  the  flowing  electrolyte  are 
governed  primarily  by  momentum  and  energy  considerations.  In  turn, 
the  shape  of  the  electrolyte  stream  determines  the  localization  of  etch- 
ing. A  stream  of  electrolyte  guided  by  wires  has  been  used  to  etch  semi- 
conductor devices.^  Surface  tension  has  an  important  influence  on  the 
free  streamlines  in  this  case, 

PROPERTIES   OF   ELECTROLYTE-SEMICONDUCTOR   BARRIERS 

The  most  distinctive  feature  of  electrolytic  etching  of  semiconductors 
is  the  occurrence  of  rectifying  barriers.  Barrier  effects  for  germanium 
will  be  described;  those  for  silicon  are  qualitatively  similar. 

The  voltage-current  curves  for  anodic  n-type  and  p-type  germanium 


ELECTROLYTIC    SHAPING   OF    GERMANIUM    AND    SILICON 


335 


[in  10  per  cent  KOH  are  shown  in  Fig.  1.  Tlie  concentration  of  KOH 

[is  not  critical  and  other  electrolytes  give  similar  results.  The  voltage 

'drop  for  the  p-type  specimen  is  small.  For  anodic  n-type  germanium, 

!  however,  the  barrier  is  in  the  reverse  or  blocking  direction  as  evidenced 

by  a  large  voltage  drop.  The  fact  that  n-type  germanium  differs  from 

p-type  germanium  only  by  very  small  amounts  of  impurities  suggests 

that  the  barrier  is  a  semiconductor  phenomenon  and  not  an  electro- 

i  chemical  one.  This  is  confirmed  by  the  light  sensitivity  of  the  n-type 

1  voltage-current  characteristic.  Fig.  2  is  a  schematic  diagram  of  the 

!  arrangement  for  obtaining  voltage-current  curves.  A  mercury-mercuric 

loxide-10  per  cent  KOH  reference  electrode  was  used  at  first,  but  a  gold 

(wire  was  found  equally  satisfactory.  At  zero  current,  a  voltage  Vo  exists 

j  between  the  germanium  and  the  reference  electrode ;  this  voltage  is  not 

[included  in  Fig.  1. 

I  The  saturation  current  Is  ,  measured  for  the  n-type  barrier  at  a 
\moderate  reverse  voltage  (see  Fig.  1),  is  plotted  as  a  function  of  tempera- 
Iture  in  Fig.  3.  The  saturation  current  increases  about  9  per  cent  per 
[degree,  just  as  for  a  germanium  p-n  junction,  which  indicates  that  the 

I 


40 


35 


30 


^25 

Lil 

O  20 


15 


10 


1 

12  OHM-CM 
n-TYPE 

/ 

DAR\<. 

1 
/ 

/ 

1 
1 
1 
1 

1 

/ 

1 
1 
1 

1 
1 

WITH  ; 
LIGHT  ^' 
1 

1 

I 
1 

1 

1 

n 

i 

1 

1 
/ 
/         P- 

FYPE 

10  20  30  40  50  60 

CURRENT   FLOW  IN  MILLIAMPERES  PER   CM^ 


Fig.  1  —  Anodic  voltage-current  characteristics  of  germanium. 


336 


THE    BELL    SYSTEM    TECHXICAL   JOURNAL,    MARCH    1956 


current  is  proportional  to  the  equilibrium  density  of  minority  carriers 
(holes).  The  same  conclusion  may  be  drawn  from  Fig.  4,  which  shows 
that  the  saturation  current  is  higher,  the  higher  the  resistivity  of  the 
n-type  germanium.  But  the  breakdown  voltages  are  variable  and  usu- 
ally much  lower  than  one  would  expect  for  planar  p-n  junctions  made, 
for  example,  by  alloying  indium  into  the  same  n-type  germanium. 

Breakdown  in  bulk  junctions  is  attributed  to  an  avalanche  multipli- 
cation of  carriers  in  high  fields.^  The  same  mechanism  may  be  responsible 
for  breakdown  of  the  germanium-electrolyte  barrier;  low  and  variable 
breakdown  voltages  may  be  caused  by  the  pits  described  below. 

The  electrolyte-germanium  barrier  exhibits  a  kind  of  current  multi- 
plication that  differs  from  high-field  multiplication  in  two  respects:  it 
occurs  at  much  lower  reverse  voltages  and  does  not  vary  much  with 
voltage.^  This  effect  can  be  demonstrated  very  simply  by  comparison 
with  a  metal-germanium  barrier,  on  the  assumption  that  the  latter  has 
a  current  multiplication  factor  of  unity.  This  assumption  is  supported 
by  experiments  which  indicate  that  current  flows  almost  entirely  by 
hole  flow,  for  good  metal-germanium  barriers. 

The  experimental  arrangement  is  indicated  in  Fig.  5(a)  and  (b).  The 
voltage-current  curves  for  an  electrolyte  barrier  and  a  plated  barrier  on 
the  same  slice  of  germanium  are  shown  in  Fig.  5(c).*  The  curves  for  the 


REFERENCE 
ELECTRODE 


CATHODE 


LIGHT 


Fig.  2  —  Arrangement  for  obtaining  voltage  current  characteristics. 


*  In  Fig.  5  the  dark  current  for  the  phited  barrier  is  much  hirger  than  can  be 
exphained  on  the  basis  of  hole  current;  it  is  even  higher  than  the  dark  current  for 
the  electrolyte  barrier,  which  should  be  at  least  1.4  times  the  hole  current.  This 
excess  dark  current  is  believed  to  be  leakage  at  the  edges  of  the  plated  area  and 
probably  does  not  affect  the  intrinsic  current  multiplication  of  the  plated  barrier 
as  a  whole. 


ELECTROLYTIC   SHAPING    OF    GERMANIUM    AND    SILICON 


337 


10 


2 
o 

a. 

01 

a. 
to 

Ui 

oc 

LU 
Q. 

5 
< 

_) 


m 
cc 
tr 

3 
U 

z 
o 

cc 

3 
(0 


•>  I 


10" 


/ 

{ 

- 

/ 

- 

1 

/ 

7 

/ 

/ 

/ 

- 

/% 

- 

/ 

/ 

/ 

/ 

n/ 

^ 

y 

i<:i 


0  10  20  30  40  50  60 

TEMPERATURE  IN  DEGREES  CENTIGRADE 

Fig.  3  —  Temperature  variation  of  the  saturation  current  of  a  barrier  between 
5.5  ohm-cm  n-type  germanium  and  10  per  cent  KOH  solution. 


illuminated  condition  were  obtained  by  shining  light  on  a  dry  face  of  a 
slice  while  the  barriers  were  on  the  other  face.  The  difference  between 
the  light  and  dark  currents  is  larger  for  the  electrolyte-germanium  bar- 
rier than  for  the  metal-germanium  barrier,  by  a  factor  of  about  1.4. 

The  transport  of  holes  through  the  slice  is  probably  not  very  different 
for  the  two  barriers.  Therefore,  a  current  multiplication  of  1.4  is  indi- 
cated for  the  electrolyte  barrier.  About  the  same  value  was  found  for 
temperatures  from  15°C  to  60°C,  KOH  concentrations  from  0.01  per 
cent  to  10  per  cent,  n-type  resistivities  of  0.2  ohm-cm  to  6  ohm-cm, 
light  currents  of  0.1  to  1.0  ma/cm^,  and  for  O.IN  indium  sulfate. 

Evidently  the  flow  of  holes  to  the  electrolyte  barrier  is  accompanied 
by  a  proportionate  return  flow  of  electrons,  which  constitutes  an  addi- 
tional electric  current.  Possible  mechanisms  for  the  creation  of  the 
electrons  will  be  discussed  in  a  forthcoming  article. 


338  THE    BELL    SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 

7 


>    4 


LU 

o 
> 


I 


0,5  1.0  1 

CURRENT  F 


5  2.0         2.5  3.0  3.5         4.0 

LOW  IN  MILLIAMPERES  PER  CM^ 


4.5 


Fig.  4.  —  Anodic  voltage -current  curves  for  various  resistivities  of  germanium. 


SCRATCHES    AND    PITTING 

The  voltage- current  curve  of  an  electrolyte-germanium   barrier  is 
very  sensitive  to  scratches.  The  curves  given  in  the  illustrations  were : 
obtained  on  material  previously  etched  smooth  in  CP-4,  a  chemical  I 
etch.*  '' 

If,  instead,  one  starts  with  a  lapped  piece  of  n-type  germanium,  the 
electrolyte-germanium  barrier  is  essentially  "ohmic;"  that  is,  the  voltage 
drop  is  small  and  proportional  to  the  current.  A  considerable  reverse 
voltage  can  be  attained  if  lapped  n-type  germanium  is  electrolytically 
etched  long  enough  to  remove  most  of  the  damaged  germanium.  How- 
ever, a  pitted  surface  results  and  the  breakdown  voltage  achieved  is 
not  as  high  as  for  a  smooth  chemically-etched  surface. 

The  depth  of  damage  introduced  by  typical  abrasive  sawing  and 
lapping  was  investigated  by  noting  the  voltage-current  curve  of  the 


Br2 


Five  parts  HNO3  ,  3  parts  48  per  cent  HF,  3  parts  glacial  acetic  acid,  ^0  P^-^t 


ELECTROLYTIC   SHAPING   OF   GERMANIUM   AND   SILICON 


339 


electrolyte-germanium  barrier  after  various  amounts  of  material  had 
been  removed  by  chemical  etching.  After  20  to  50  microns  had  been  re- 
moved, further  chemical  etching  produced  no  change  in  the  barrier 
characteristic.  This  amount  of  material  had  to  be  removed  even  if  the 
lapping  was  followed  by  polishing  to  a  mirror  finish.  The  voltage-current 
curve  of  the  electrolyte-germanium  barrier  will  reveal  localized  damage. 
On  the  other  hand,  the  photomagnetoelectric  (PME)  measurement  of 


I 

-< — 

REFERENCE 
ELECTRODE 

CATHODE- 

-- 

-^ 

■< 

■y 

GLASS   TUBING 

CEMENTED 

TO  Ge 

E 

LECTROLYT 

z         i 

N-Ge 

■^ 

1 

1 

1 
1 

<rri> 


(a) 


ELECTROPLATED 
INDIUM 


METAL  TO  N-Ge 
CONTACT 

ELECTROLYTE   TO 
N-Ge    BARRIER 


(c) 


0  2  4  6 

CURRENT,  I,  IN   MILLIAMPERES 

PER    CM  2 


Fig.  5  —  Determination  of  the  current  multiplication  of  the  barrier  between 
6  ohm-cm  n-type  germanium  and  an  electrolyte. 


340  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    1956 


Fig.  6  —  Electrolytic  etch  pits  on  two  sides  of  0.02-inch  slice  of  n-type  germa- 
nium. Half  of  the  slice  was  in  contact  with  the  electrolyte. 

surface  recombination  velocity  gives  an  evaluation  of  the  average  con- 
dition of  the  surface.  A  variation  of  the  PME  method  has  been  used 
to  study  the  depth  of  abrasion  damage;  the  damage  revealed  by  this 
method  extends  only  to  a  depth  comparable  to  the  abrasive  size. 

A  scratch  is  sufficient  to  start  a  pit  that  increases  in  size  without  limit 
if  anodic  etching  is  prolonged.  However,  a  scratch  is  not  necessary.  Pits 
are  formed  even  when  one  starts  with  a  smooth  surface  produced  by 
chemical  etching.  A  drop  in  the  breakdown  voltage  of  the  barrier  is 
noticed  when  one  or  more  pits  form.  The  breakdown  voltage  can  be 
restored  by  masking  the  pits  with  polystyrene  cement. 

Evidence  that  the  spontaneous  pits  are  caused  by  some  features  of 
the  crystal,  itself,  was  obtained  from  an  experiment  on  single-crystal 
n-type  germanium  made  by  an  early  version  of  the  zone-leveling  process. 
A  slice  of  this  material  was  electrolytically  etched  on  both  sides,  after 
preliminary  chemical  etching.  Photographs  of  the  two  sides  of  the  slice 
are  shown  in  Fig.  6.  Only  half  of  the  slice  was  immersed  in  the  electro- 
lyte. The  electrolytic  etch  pits  are  concentrated  in  certain  regions  of 
the  slice  —  the  same  general  regions  on  both  sides  of  the  slice.  It  is 
interesting  that  radioautographs  and  resistivity  measurements  indicate 
high  donor  concentrations  in  these  regions.  Improvements,  including 
more  intensive  stirring,  were  made  in  the  zone-leveling  process,  and  the 
electrolytic  etch  pit  distribution  and  the  donor  radioautographs  have 
been  much  more  uniform  for  subsequent  material. 

Several  pits  on  a  (100)  face  are  shown  in  Fig.  7.  The  pits  grow  most 
rapidly  in  (100)  directions  and  give  the  spiked  effect  seen  in  the  illustra- 
tion. Toiler  prolonged  etching,  the  spikes  and  their  branches  form  a  com- 
plex network  of  caverns  beneath  the  surface  of  the  germanium. 

High-field  carrier  generation  may  be  responsible  for  pitting.  A  locally 


ELECTROLYTIC   SHAPING    OF    GERMAXIUM   AND    SILICON 


341 


Fig.  7  —  Electrolytic  etch  pits  on  n-type  germanium. 

high  donor  concentration  would  favor  breakdown,  as  would  any  con- 
cavity of  the  germanium  surface  (which  would  cause  a  higher  field  for 
a  given  voltage) .  Very  high  fields  must  occur  at  the  points  of  spikes  such 

jas  those  shown  in  Fig.  7.  The  continued  growth  of  the  spikes  is  thus 
favored  by  their  geometry. 

Microscopic  etch  pits  arising  from  chemical  etching  have  been  corre- 

;lated  with  the  edge  dislocations  of  small-angle  grain  boundaries.     A 

I  specimen  of  n-type  germanium  with  chemical  etch  pits  was  photomicro- 
graphed  and  then  etched  electrolytically.  The  etch  pits  produced  elec- 
trolytically  could  not  be  correlated  with  the  chemical  etch  pits,  most 
of  which  were  still  visible  and  essentially  unchanged  in  appearance. 
Also,  no  correlation  could  be  found  between  either  kind  of  etch  pit  and 
the  locations  at  which  copper  crystallites  formed  upon  immersion  in  a 
copper  sulfate  solution.  Microscopic  electrolytic  etch  pits  at  dislocations 

j  in  p-type  germanium  have  been  reported  in  a  recent  paper  that  also 
I  mentions  the  deep  pits  produced  on  n-type  germanium.^* 
y     Electrolytic  etch  pits  are  observed  on  n-type  and  high-resistivity 
silicon.  These  etch  pits  are  more  nearly  round  than  those  produced  in 
germanium. 

In  spite  of  the  pitting  phenomenon,  electrolytic  etching  is  success- 


342 


THE   BELL   SYSTEM   TECHNICAL  JOURNAL,   MARCH    1956 


I 

fully  used  in  the  fabrication  of  devices  involving  n-type  semiconductors. 
Pitting  can  be  reduced  relative  to  "normal"  uniform  etching  by  any 
agency  that  increases  the  concentration  of  holes  in  the  semiconductor. 
Thus,  elevated  temperatures,  flooding  with  light,  and  injection  of  holes 
by  an  emitter  all  favor  smooth  etching. 


SHAPING   BY   MEANS   OF   INJECTED    CARRIERS 


I 


Hole-electron  pairs  are  produced  when  light  is  absorbed  by  semi- 
conductors. Light  of  short  wavelength  is  absorbed  in  a  short  distance, 
while  long  wavelength  light  causes  generation  at  considerable  depths. 
The  holes  created  by  the  light  move  by  diffusion  and  drift  and  increase 
the  current  flow  through  an  anodic  electrolyte-germanium  barrier  at 
whatever  point  they  happen  to  encounter  the  barrier.  In  general,  more 
holes  will  diffuse  to  a  barrier,  the  nearer  the  barrier  is  to  the  point  at 
which  the  holes  are  created.  For  n-type  semiconductors,  the  current 
due  to  the  light  can  be  orders  of  magnitude  greater  than  the  dark  cur- 
rent, so  that  the  shape  resulting  from  etching  is  almost  entirely  deter- 
mined by  the  light.  As  shown  in  Fig.  3,  the  dark  current  can  be  made 
very  small  by  lowering  the  temperature. 

An  example  of  the  shaping  that  can  be  done  with  light  is  shown  in 
Fig.  8.  A  spot  of  light  impinges  on  one  side  of  a  wafer  of  n-type  germanium 
or  silicon.  The  semiconductor  is  made  anodic  with  respect  to  an  etching 
electrolyte.  Accurately  concentric  dimples  are  produced  on  both  sides  of 
the  wafer.  Two  mechanisms  operate  to  transmit  the  effect  to  the  oppo- 
site side.  One  is  that  some  of  the  light  may  penetrate  deeply  before 
generating  a  hole-electron  pair.  The  other  is  that  a  fraction  of  the  car- 
riers generated  near  the  first  surface  will  diffuse  to  the  opposite  side. 
By  varying  the  spectral  content  of  the  light  and  the  depth  within  the  \ 


\ 


-n-TYPE    SEMICONDUCTOR 


LIGHT 


I  I 


Fig.  8  —  Double  dimpling  with  light. 


ELECTROLYTIC   SHAPING   OF    GERMANIUM   AND   SILICON 


343 


wafer  at  which  the  light  is  focused,  one  can  produce  dimples  with  a  vari- 
,'ety  of  shapes  and  relative  sizes. 

I  It  is  obvious  that  the  double-dimpled  wafer  of  Fig.  8  is  desirable  for 
{the  production  of  p-n-p  alloy  transistors.  For  such  use,  one  of  the  most 
[important  dimensions  is  the  thickness  remaining  between  the  bottoms 
of  the  two  dimples.  As  has  been  mentioned  in  connection  with  the  jet- 
I  etching  process,  a  convenient  way  of  monitoring  this  thickness  to  de- 
Itermine  the  endpoint  of  etching  is  to  note  the  transmission  of  light  of 
[suitable  wavelength.^  There  is,  however,  a  control  method  that  is  itself 
[automatic.  It  is  based  on  the  fact  that  at  a  reverse-biased  p-n  junction 
[Or  electrolyte-semiconductor  barrier  there  is  a  space-charge  region  that 
is  practically  free  of  carriers.  When  the  specimen  thickness  is  reduced 
so  that  space-charge  regions  extend  clear  through  it,  current  ceases  to 
flow  and  etching  stops  in  the  thin  regions,  as  long  as  thermally  or  op- 
tically generated  carriers  can  be  neglected.  However,  more  pitting  is  to 
be  expected  in  this  method  than  when  etching  is  conducted  in  the  pres- 
ence of  an  excess  of  injected  carriers. 

A  p-n  junction  is  a  means  of  injecting  holes  into  n-type  semiconduc- 
tors and  is  the  basis  of  another  method  of  dimpling,  shown  in  Fig.  9. 
The  p-n  junction  can  be  made  by  an  alloying  process  such  as  bonding 
an  acceptor-doped  gold  wire  to  germanium.  The  ohmic  contact  can  be 
made  by  bonding  a  donor-doped  gold  wire  and  permits  the  injection  of 
a  greater  excess  of  holes  than  would  be  possible  if  the  current  through 
the  p-n  junction  were  exactly  equal  to  the  etching  current.  Dimpling 
without  the  ohmic  contact  has  been  reported.^ 


14 


OHMIC    CONTACT 


p-n   JUNCTION 


Fig.  9  —  Dimpling  with  carriers  injected  by  a  p-n  junction. 


344 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


CONTROL   BY   OHMIC    CONDUCTION 

The  carrier-injection  shaping  techniques  work  very  well  for  n-typei 
material.  It  is  also  possible  to  inject  a  significant  number  of  holes  intos 
rather  high  resistivity  p-type  material.  But  what  can  be  done  about: 
p-type  material  in  general,  short  of  developing  cathodic  etches?  ] 

The  ohmic  resistivity  of  p-type  material  can  be  used  as  shown  in  Fig.!^ 
10.  More  etching  currect  flows  through  surfaces  near  the  small  contact 
than  through  more  remote  surfaces.  A  substantial  dimpling  effect  is 
observed  when  the  semiconductor  resistivity  is  equal  to  the  electrolyte 
resistivity,  but  improved  dimpling  is  obtained  on  higher  resistivity 
semiconductor.  This  result  is  just  what  one  might  expect.  But  the  math- 
ematical solution  for  ohmic  flow  from  a  point  source  some  distance  from 
a  planar  boundary  between  semi-infinite  materials  of  different  conduc- 
tivities shows  that  the  current  density  distribution  does  not  depend  on 
the  conductivities.  An  important  factor  omitted  in  the  mathematical 
solution  is  the  small  but  significant  barrier  voltage,  consisting  largely  of 
electrochemical  polarization  in  the  electrolyte.  The  barrier  voltage  is; 
approximately  proportional  to  the  logarithm  of  the  current  density; 
while  the  ohmic  voltage  drops  are  proportional  to  current  density.  Thus,- 
high  current  favors  localization. 

ELECTROLYTES   FOR   ETCHING   GERMANIUM   AND    SILICON  » 

The  electrolyte  usually  has  two  functions  in  the  electrolytic  etching 
of  an  oxidizable  substance.  First,  it  must  conduct  the  current  necessary 
for  the  oxidation.  Second,  it  must  somehow  effect  removal  of  the  oxida- 
tion product  from  the  surface  of  the  material  being  etched. 

The  usefulness  of  an  electrolytic  etch  depends  upon  one  or  both  of: 


ANY    CONTACT, 
PREFERABLY  OHMIC 


^//yyyy//y/y/y////////y////y/////yyyyyyyyyyy7^ 


Fig.  10  —  Dimpling  by  ohmic  conduction. 


ELECTROLYTIC    SHAPING    OF    GERMANIUM   AND    SILICON  345 

the  following  situations  —  the  electrolytic  process  accomplishes  a  reac- 
tion that  cannot  be  achieved  as  conveniently  in  any  other  way  or  it 
permits  greater  control  to  be  exercised  over  the  reaction.  Accordingly, 
chemical  attack  by  the  chosen  electrolyte  must  be  slight  relative  to  the 
electrochemical  etching. 

A  smooth  surface  is  probably  desirable  in  the  neighborhood  of  a  p-n 
junction,  to  avoid  field  concentrations  and  lowering  of  breakdown 
voltage.  Therefore,  a  tentative  requirement  for  an  electrolyte  is  the 
production  of  a  smooth,  shiny  surface  on  the  p-type  semiconductor.  Such 

\  an  electrolyte  will  give  a  shiny  but  possibly  pitted  surface  on  n-type 

j  specimens  of  the  same  semiconductor. 

The  effective  valence  of  a  material  being  electrolytically  etched  is 

;  defined  as  the  number  of  electrons  that  traverse  the  circuit  divided  by 
the  number  of  atoms  of  material  removed.  (The  amount  of  material 

!  removed  was  determined  by  weighing  in  the  experiments  to  be  described.) 
If  the  effective  valence  turns  out  to  be  less  than  the  valence  one  might 
predict  from  the  chemistry  of  stable  compounds,  the  etching  is  sometimes 
said  to  be  "more  than  100  per  cent  efficient."  Since  the  anode  reactions 
in  electrolytic  etching  may  involve  unstable  intermediate  compounds 
and  competing  reactions,  one  need  not  be  surprised  at  low  or  fractional 
effective  valences. 

Germanium  can  be  etched  in  many  aqueous  electrolytes.  A  valence  of 
almost  exactly  4  is  found.  That  is,  4  electrons  flow  through  the  circuit 
for  each  atom  of  germanium  removed.  For  accurate  valence  measure- 
ments, it  is  advisable  to  exclude  oxygen  by  using  a  nitrogen  atmosphere. 
Potassium  hydroxide,  indium  sulfate,  and  sodium  chloride  solutions  are 
among  those  that  have  been  used.  Sulfuric  acid  solutions  are  prone  to 

)  yield  an  orange-red  deposit  which  may  be  a  suboxide  of  germanium/* 

I  Similar  orange  deposits  are  infrequently  encountered  with  potassium 

I  hydroxide. 

Hydrochloric  acid  solutions  are  satisfactoiy  electrolytes.  The  reaction 

I  product  is  removed  in  an  unusual  manner  when  the  electrolyte  is  about 
2N  hydrochloric  acid.  Small  droplets  of  a  clear  liquid  fall  from  the  etched 
regions.  These  droplets  may  be  germanium  tetrachloride,  which  is  denser 
than  the  electrolyte.  They  turn  brown  after  a  few  seconds,  perhaps  be- 
cause of  hydrolysis  of  the  tetrachloride. 

Etching  of  germanium  in  sixteen  different  aqueous  electroplating 
electrolytes  has  been  mentioned.  Germanium  can  also  be  etched  in  the 
partly  organic  electrolytes  described  below  for  silicon. 

One  would  expect  that  silicon  could  be  etched  by  making  it  the  anode 
in  a  cell  with  an  aqueous  hydrofluoric  acid  electrolyte.  The  seemingly 


346  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956  | 

) 

likely  oxidation  product,  silicon  dioxide,  should  react  with  the  hydro-! 
fluoric  acid  to  give  silicon  tetrafluoride,  which  could  escape  as  a  gas.  In 
fact,  a  gas  is  formed  at  the  anode  and  the  silicon  loses  weight.  But  the 
gas  is  hydrogen  and  an  effective  valence  of  2.0  ±  0.2  (individual  deter- 
minations ranged  from  1.3  to  2.7)  was  found  instead  of  the  value  4  that  i 
might  have  been  expected.  The  quantity  of  hydrogen  evolved  is  con- 
sistent with  the  formal  reaction 


Si  —>  Si"*"'"  +  me  (electrochemical  oxidation) 

Si+™  +  (4-to)H+  -^  Si+'  +  Vz  (4-m)H2  (chemical  oxidation) 


where  m  is  about  two.  The  experiments  were  done  in  24  per  cent  to  48 
per  cent  aqueous  solutions  of  HF  at  current  densities  up  to  0.5  amp/cm^. 

The  suggestion  that  the  electrochemical  oxidation  precedes  the  chemi- 
cal oxidation  is  supported  by  the  appearance  and  behavior  of  the  etched 
surfaces.  Instead  of  being  shiny,  the  surfaces  have  a  matte  black,  brown, 
or  red  deposit. 

At  40 X  magnification,  the  deposit  appears  to  consist  of  flakes  of  a; 
resinous  material,  tentatively  supposed  to  be  a  silicon  suboxide.  A  re- 
markable reaction  can  be  demonstrated  if  the  silicon  is  rinsed  briefly  in 
water  and  alcohol  after  the  electrolytic  etch,  dried,  and  stored  in  air  for 
as  long  as  a  year.  Upon  reimmersing  this  silicon  in  water,  one  can  observe 
the  liberation  of  gas  bubbles  at  its  surface.  This  gas  is  presumed  to  be 
hydrogen.  To  initiate  the  reaction  it  is  sometimes  necessary  to  dip  the 
specimen  first  in  alcohol,  as  water  may  otherwise  not  wet  it.  The  speci- 
mens also  liberate  hydrogen  from  alcohol  and  even  from  toluene. 

Thus,    chemical    oxidation    can   follow   electrolytic    oxidation.   But 
chemical  oxidation  does  not  proceed  at  a  significant  rate  before  thei 
current  is  turned  on. 

Smooth,  shiny  electrolytic  etching  of  p-type  silicon  has  been  obtained; 
with  mixtures  of  hydrofluoric  acid  and  organic  hydroxyl  compounds,; 
such  as  alcohols,  glycols,  and  glycerine.  These  mixtures  may  be  an- 
hydrous or  may  contain  as  much  as  90  per  cent  water.  The  organic 
additives  tend  to  minimize  the  chemical  oxidation  of  the  silicon.  They; 
also  permit  etching  at  temperatures  below  the  freezing  point  of  aqueous 
solutions.  They  lower  the  conductivity  of  the  electrolyte. 

For  a  given  electrolyte  composition,  there  is  a  threshold  current 
density,  usually  between  0.01  and  0.1  amps/cm  ,  for  smooth  etching.; 
Lower  current  densities  give  black  or  red  surfaces  with  the  same  hy- 
drogen-liberating capabilities  as  those  obtained  in  aqueous  hydrofluoric 
acid. 


ELECTROLYTIC   SHAPING   OF   GERMANIUM   AND   SILICON  347 

In  general,  smooth  etching  of  siHcon  seems  to  result  when  the  effective 
valence  is  nearly  4  and  there  is  little  anodic  evolution  of  gas.  The  elec- 
I  trical  properties  of  the  smooth  surface  appear  to  be  equivalent  to  those 
!  of  smooth  silicon  surfaces  produced  by  chemical  etching  in  mixtures  of 
i  nitric  and  hydrofluoric  acids.  On  the  other  hand,  the  reactive  surface 
[produced  at  a  valence  of  about  2,  with  anodic  hydrogen  evolution,  is 
I  capable  of  practically  shorting-out  a  silicon  p-n  junction.  The  electrical 
j  properties  of  this  surface  tend  to  change  upon  standing  in  air. 

ACKNOWLEDGEMENTS 

Most  of  the  experiments  mentioned  in  this  paper  were  carried  out  by 
my  wife,  Ingeborg.  An  exception  is  the  double-dimpling  of  germanium 
by  light,  which  was  done  by  T.  C.  Hall.  The  dimpling  procedures  of 
Figs.  9  and  10  are  based  on  suggestions  by  J.  M.  Early.  The  effect  of 
light  upon  electrolytic  etching  was  called  to  my  attention  by  0.  Loosme. 
W.  G.  Pfann  provided  the  germanium  crystals  grown  with  different 
degrees  of  stirring. 

REFERENCES 

1.  J.  F.  Barry,  I.R.E.-A.I.E.E.  Semiconductor  Device  Research  Conference, 

Philadelphia,  June,  1955. 

2.  A.  Uhlir,  Jr.,  Rev.  Sci.  Inst.,  26,  pp.  965-968,  1955. 

3.  W.  E.  Bailey,  U.  S.  Patent  No.  1,416,  929,  May  23,  1922. 

4.  Bradley,  et  al.  Proc.  I.R.E.,  24,  pp.  1702-1720,  1953. 

5.  M.  V.  Sullivan  and  J.  H.  Eigler,  to  be  published. 

6.  S.  L.  Miller,  Phys.  Rev.  99,  p.  1234,  1955. 

7.  W.  H.  Brattain  and  C.  G.  B.  Garrett,  B.S.T.J.,  34,  pp.  129-176,  1955. 

8.  E.  H.  Borneman,  R.  F.  Schwarz,  and  J.  J.  Stickler,  J.  Appl.  Phvs.,  26,   pp. 

1021-1029,  1955. 

9.  D.  R.  Turner,  to  be  submitted  to  the  Journal  of  the  Electrochemical  Society. 

10.  R.  D.  Heidenreich,  U.  S.  Patent  No.  2,619,414,  Nov.  25,  1952. 

11.  T.  S.  Moss,  L.  Pincherle,  A.  M.  Woodward,  Proc.  Phys.  Soc.  London,  66B, 

p.  743,  1953. 

12.  T.  M.  Buck  and  F.  S.  McKim,  Cincinnati  Meeting  of  the  Electrochemical 

Society,  Mav,  1955. 

13.  F.  L.  Vogel,  W.  G.  Pfann,  H.  E.  Corey,  and  E.  E.  Thomas,  Phys.  Rev.,  90, 

p.  489,  1953. 

14.  S.  G.  Ellis,  Phys.  Rev.,  100,  pp.  1140-1141,  1955. 

15.  Electronics,  27,  No.  5,  p.  194,  May,  1954. 

16.  F.  Jirsa,  Z.  f.  Anorg.  u.  AUgemeine  Chem.,  Bd.  268,  p.  84,  1952. 


\ 


A  Large  Signal  Theory  of  Traveling 
Wave  Amplifiers 

Including  the  Effects  of  Space  Charge  and  Finite 
Coupling  Between  the  Beam  and  the  Circuit 

By  PING  KING  TIEN 

Manuscript  received  October  11,  1955) 

The  non-linear  behavior  of  the  traveling-wave  amplifier  is  calculated  in 
this  paper  by  numericalhj  integrating  the  motion  of  the  electrons  in  the 
presence  of  the  circuit  and  the  space  charge  fields.  The  calculation  extends 
the  earlier  work  by  Nordsieck  and  the  srnall-C  theory  by  Tien,  Walker  and 
Wolontis,  to  include  the  space  charge  repulsion  between  the  electrons  and 
the  effect  of  a  finite  coupling  between  the  circuit  and  the  electron  beam.  It 
however  differs  from  Poulter's  and  Rowers  works  in  the  methods  of  calcu- 
lating the  space  charge  and  the  effect  of  the  backward  wave. 

The  numerical  work  was  done  using  701 -type  I.B.M.  equipment.  Re- 
sults of  calcidation  covering  QC  from  0.1  to  0.4,  b  from  0.46  to  2.56  and  k 
from  1.25  to  2.50,  indicate  that  the  saturation  efficiency  varies  between 
23  per  cent  and  37  per  cent  for  C  equal  to  0.1  and  between  33  per  cent  and 
Jf.0  per  cent  for  C  equal  to  0.15.  The  voltage  and  the  phase  of  the  circuit  wave, 
the  velocity  spread  of  the  electrons  and  the  fundamental  component  of  the 
charge-density  modidation  are  either  tabulated  or  presented  in  curves.  A 
method  of  calculating  the  backward  wave  is  provided  and  its  effect  fully 
discussed. 

1.   INTRODUCTION 

Theoretical  evaluation  of  the  maximum  efficiency  attainable  in  a 
traveling-wave  amplifier  requires  an  understanding  of  the  non-linear 
behavior  of  the  device  at  various  working  conditions.  The  problem  has 
been  approached  in  many  ways.  Pierce/  and  later  Hess,^  and  Birdsalf 
and  Caldwell  investigated  the  efficiency  or  the  output  power,  using  cer- 
tain specific  assumptions  about  the  highly  bunched  electron  beam.  They 
either  assume  a  beam  in  the  form  of  short  pulses  of  electrons,  or,  specify 

349 


350  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 

an  optimum  ratio  of  the  fundamental  component  of  convection  current 
to  the  average  or  d-c  current.  The  method,  although  an  abstract  one, 
generally  gives  the  right  order  of  the  magnitude.  When  the  usual  wave 
concept  fails  for  a  beam  in  which  overtaking  of  the  electrons  arises,  we 
may  either  overlook  effects  from  overtaking,  or,  using  the  Boltzman's 
transport  equation  search  for  solutions  in  series  form.  This  attack  has 
been  pursued  by  Parzen  and  Kiel,  although  their  work  is  far  from  com- 
plete. The  most  satisfying  approach  to  date  is  Nordsieck's  analysis.' 
Nordsieck  followed  a  typical  set  of  "electrons"  and  calculated  their 
velocities  and  positions  by  numerically  integrating  a  set  of  equations  of 
motion.  Poulter  has  extended  Nordsieck  equations  to  include  space 
charge,  finite  C  and  circuit  loss,  although  he  has  not  perfectly  taken  into 
account  the  space  charge  and  the  backward  wave.  Recently  Tien, 
Walker,  and  Wolontis  have  published  a  small  C  theory  in  which  "elec- 
trons" are  considered  in  the  form  of  uniformly  charged  discs  and  the 
space  charge  field  is  calculated  by  computing  the  force  exerted  on  one 
disc  by  the  others.  Results  extended  to  finite  C,  have  been  reported  by 
Rowe,^*^  and  also  by  Tien  and  Walker.^^  Rowe,  using  a  space  charge 
expression  similar  to  Poulter's,  computed  the  space  charge  field  based  on 
the  electron  distribution  in  time  instead  of  the  distribution  in  space.  This 
may  lead  to  appreciable  error  in  his  space  charge  term,  although  its 
influence  on  the  final  results  cannot  be  easily  evaluated. 

In  the  present  analysis,  we  shall  adopt  the  model  described  by  Tien, 
Walker  and  Wolontis,  but  wish  to  add  to  it  the  effect  of  a  finite  beam  to 
circuit  coupling.  A  space  charge  expression  is  derived  taking  into  account 
the  fact  that  the  a-c  velocities  of  the  electrons  are  no  longer  small  com- 
pared with  the  average  velocity.  Equations  are  rewritten  to  retain  terms 
involving  C.  As  the  backward  wave  becomes  appreciable  when  C  in- 
creases, a  method  of  calculating  the  backward  wave  is  provided  and  the 
effect  of  the  backward  wave  is  studied.  Finally,  results  of  the  calculation 
covering  useful  ranges  of  design  and  operating  parameters  are  presented 
and  analyzed. 

2.   ASSUMPTIONS 

To  recapitulate,  the  major  assumptions  which  we  have  made  are: 

1.  The  problem  is  considered  to  be  one  dimensional,  in  the  sense  that 
the  transverse  motions  of  the  electrons  are  prohibited,  and  the  current, 
velocity,  and  fields,  are  functions  only  of  the  distance  along  the  tube  and 
of  the  time. 

2.  Only  the  fundamental  component  of  the  current  excites  waves  on 
the  circuit. 


A   LARGE    SIGNAL   THEORY    OF   TRAVELING-WAVE   AMPLIFIERS       351 

3.  The  space  charge  field  is  computed  from  a  model  in  which  the 
helix  is  replaced  by  a  conducting  cylinder,  and  electrons  are  uniformly 
charged  discs.  The  discs  are  infinitely  thin,  concentric  with  the  helix  and 
have  a  radius  equal  to  the  beam  radius. 

4.  The  circuit  is  lossfree. 

These  are  just  the  assumptions  of  the  Tien-Walker-Wolontis  model. 
In  addition,  we  shall  assume  a  small  signal  applied  at  the  input  end  of  a 
long  tube,  where  the  beam  entered  unmodulated.  What  we  are  looking 
for  are  therefore  the  characteristics  of  the  tube  beyond  the  point  at  which 
the  device  begins  to  act  non-linearly.  Let  us  imagine  a  flow  of  electron 
discs.  The  motions  of  the  discs  are  computed  from  the  circuit  and  the 
space  charge  fields  by  the  familiar  Newton's  force  equation.  The  elec- 
trons, in  turn,  excite  waves  on  the  circuit  according  to  the  circuit  equa- 
tion derived  either  from  Brillouin's  model^  or  from  Pierce's  equivalent 
circuit.  The  force  equation,  the  circuit  equation,  and  the  equation  of 
conservation  of  charge  in  kinematics,  are  the  three  basic  equations 
from  which  the  theory  is  derived. 

3.   FORWARD   AND   BACKWARD   WAVES 

In  the  traveling-wave  amplifier,  the  beam  excites  forward  and  back- 
ward waves  on  the  circuit.  (We  mean  by  "forward"  wave,  the  wave 
which  propagates  in  the  direction  of  the  electron  flow,  and  by  "back- 
ward" wave,  the  wave  which  propagates  in  the  opposite  direction.) 
Because  of  phase  cancellation,  the  energy  associated  with  the  backward 
wave  is  small,  but  increases  with  the  beam  to  circuit  coupling.  It  is  there- 
fore important  to  compute  it  accurately.  In  the  first  place,  the  waves  on 
the  circuit  must  satisfy  the  circuit  equation 

dH^(z,t)  2d'V{z,t)  „    d'p^iz,   t)  ,v 

Here,  V  is  the  total  voltage  of  the  waves.  Vo  and  Zo  are  respectively  the 
phase  velocity  and  the  impedance  of  the  cold  circuit,  z  is  the  distance 
along  the  tube  and  t,  the  time,  p^  is  the  fundamental  component  of  the 
linear  charge  density.  V  and  p„  are  functions  of  z  and  /.  The  complete 
solution  of  (1)  is  in  the  form 

Viz)  =  Cre'^''  +  (726  "^"^ 

+  e         —-y—  J    e  "  po,{^)  dz  ^2) 

+  e  "   —^  j     e       p^{z)  dz 


352  THE   BELL   SYSTEM   TECHNICAL  JOURNAL,    MARCH    1956 

where  the  common  factor  e^"'  is  omitted.  To  =  j{co/vo),  j  =  \/—  1  and  w 
is  the  angular  frequency.  Ci  and  C2  are  arbitrary  constants  which  will 
be  determined  by  the  boundary  conditions  at  the  both  ends  of  the  beam. 
The  first  two  terms  are  the  solutions  of  the  homogeneous  equation  (or 
the  complementary  functions)  and  are  just  the  cold  circuit  waves.  The 
third  and  the  fourth  terms  are  functions  of  electron  charge  density  and 
are  the  particular  solution  of  the  equation. 

Let  us  consider  a  long  traveling-wave  tube  in  which  the  beam  starts 
from  z  =  0  and  ends  at  2;  =  D.  The  motion  of  electrons  observed  at  any 
particular  position  is  periodic  in  time,  though  it  varies  from  point  to 
point  along  the  beam.  To  simplify  the  picture,  we  may  divide  the  beam 
along  the  tube  into  small  sections  and  consider  each  of  them  as  a  current 
element  uniform  in  z  and  periodic  in  time.  Each  section  of  beam,  or  each 
current  element  excites  on  the  circuit  a  pair  of  waves  equal  in  ampli- 
tudes, one  propagating  toward  the  right  (i.e.,  forward)  and  the  other, 
toward  the  left.  One  may  in  fact  imagine  that  these  are  trains  of  waves 
supported  by  the  periodic  motion  of  the  electrons  in  that  section  of  the 
beam.  Obviously,  a  superposition  of  these  waves  excited  by  the  whole 
beam  gives  the  actual  electromagnetic  field  distribution  on  the  circuit. 
One  may  thus  compute  the  forward  traveling  wave  at  z  by  summing  all 
the  waves  at  z  which  come  from  the  left.  Stated  more  specifically,  the 
forward  traveling  energy  at  z  is  contributed  by  the  waves  excited  by  the 
current  elements  at  the  left  of  the  point  z.  Similarly  the  backward  travel- 
ing energy,  (or  the  backward  wave)  at  z  is  contributed  by  the  waves 
excited  by  the  current  elements  at  the  right  of  the  point  z.  It  follows 
obviously  from  this  picture  that  there  is  no  forward  wave  at  2  =  0 
(except  one  corresponding  to  the  input  signal),  and  no  backward  wave 
at  2  =  D.  (This  implies  that  the  output  circuit  is  matched.)  With  these 
boundary  conditions,  (1)  is  reduced  to 


z)  =  Finput  e     "    +  e     °   — -—  /     e  "  po,{z) 

Z      Jo 


dz 


+  /-^J  e-%.(.) 


(3) 


dz 


Equations  (2)  and  (3)  have  been  obtained  by  Poulter.^  The  first  term  of 
(3)  is  the  wave  induced  by  the  input  signal.  It  propagates  as  though  the  ; 
beam  were  not  present.  The  second  term  is  the  voltage  at  z  contributed 
by  the  charges  between  2  =  0  and  2  =  2.  It  is  just  the  voltage  of  the 
forward  wave  described  earlier.  Similarly  the  third  term  which  is  the 
voltage  at  2  contributed  by  the  charges  between  z  =  z  and  2  =  D  is  the 
voltage  of  the  backward  wave  at  the  point  2.  Denote  F  and  B  respec- 


A   LARGE   SIGNAL   THEORY    OF    TRAVELING-WAVE    AMPLIFIERS       353 

tively  the  voltages  of  the  forward  and  the  backward  waves,  we  have 
F{z)  =  Fi„put  e-'"^  +  e-^»^  ^«  r  e'^'  p^z)  dz  (4) 

Z         Jo 

Biz)  =  e^-  ^°  £  e-^-p„(e)  dz  (5) 

It  can  be  shown  by  direct  substitution  that  F  and  B  satisfy  respectively 
the  differential  equations 


dz              Vo        dt               2         (9^ 

(6) 

dB(z,  t)      1  a5(2,  0         Zo  ap„(2, 0 

(92              1^0        di                    2         dt 

(7) 

We  put  (4)  and  (5)  in  the  form  of  (6)  and  (7)  simply  because  the  differ- 
ential equations  are  easier  to  manipulate  than  the  integral  equations. 
In  fact,  we  should  start  the  analysis  from  (6)  and  (7)  if  it  were  not  for  a 
physical  picture  useful  to  the  understanding  of  the  problem.  Equations 
(6)  and  (7)  have  the  advantage  of  not  being  restricted  by  the  boundary 
conditions  at  2;  =  0  and  D,  which  we  have  just  imposed  to  derive  (4) 
and  (5).  Actually,  we  can  derive  (6)  and  (7)  directly  from  the  Brillouin 
model  in  the  following  manner.  Suppose  Y,  I  and  Zo  are  respectively 
the  voltage,  current  and  the  characteristic  impedance  of  a  transmission 
line  system  in  the  usual  sense.  (V  +  /Zo)  must  then  be  the  forward  wave 
and  {V  —  IZo)  must  be  the  backward  wave.  If  we  substituted  F  and  B 
in  these  forms  into  (1)  of  the  Brillouin' s  paper,^^  we  should  obtain  exactly 
(6)  and  (7). 

It  is  obvious  that  the  first  and  third  terms  of  (2)  are  respectively  the 
complementary  function  and  the  particular  solution  of  (6),  and  similarly 
the  second  and  the  fourth  terms  of  (2)  are  respectively  the  comple- 
mentary function  and  the  particular  solution  of  (7).  From  now  on,  we 
shall  overlook  the  complementary  functions  which  are  far  from  syn- 
chronism with  the  beam  and  are  only  useful  in  matching  the  boundary 
conditions.  It  is  the  particular  solutions  which  act  directly  on  the  elec- 
tron motion.  With  these  in  mind,  it  is  convenient  to  put  F  and  B  in  the 
form 

Fiz,  t)  =  -j~  [aiiij)  cos  <p  -  aiiy)  sin  ^]  (8) 

B{z,  t)  =  -^  [hiiy)  cos  ip  -  h^iy)  sin  9?]  (9) 


354  THE   BELL   SYSTEM   TECHNICAL  JOURNAL,    MARCH    1956 

where  ai(y),  02(2/),  hi(y)  and  62(2/)  are  functions  of  y.  y  and  <p  are  inde- 
pendent variables  and  have  been  used  by  Nordsieck  to  replace  the  vari- 
ables, z  and  t,  such  as 

y  =  C  —  Z 

(f   =   w  [  —    —    t  ] 

\Vo         / 

Here  as  defined  earlier,  I'o  is  the  phase  velocity  of  the  cold  circuit  and  Vq 
the  average  velocity  of  the  electrons.  They  are  related  by  the  parameter 
h  defined  by  Pierce  as 

Uo  1 


vo        (1  -  hC) 
C  is  the  gain  parameter  also  defined  by  Pierce, 

^3  _  ZqIo 

in  which,  Vo  and  7o  are  respectively  the  beam  voltage  and  current. 
Adding  (6)  to  (7),  we  obtain  an  important  relation  between  F  and  B, 
that  is, 

dFjz,  t)  _^  1_  dF{z,  t)  ^       dBjz,  t)  _j_  l_  dBjz,  t)  ^^Q^ 

dz  Vo        dt  dz  Vo        dt 

Substituting  (8)  and  (9)  into  (10)  and  carrying  out  some  algebraic 
manipulation,  we  obtain 

'"'^^  =  "2(1  +  bC)  I  ^'^^'>  +  "'-^"^^ 

(11) 

"'^^'^  =  2(1  +  bC)  ly  '"'^^^  +  '"^^^' 


or 


B{z,  t)  = 


ZqIo  C 

dMy)  +  bM)  ,„,  ^  +  diaM+  b.(,j))  ^.^    - 
dy  dy 


[ 


For  better  understanding  of  the  problem,  we  shall  first  solve  (12a)  ap- 
proximately. Assuming  for  the  moment  that  hiiy)  and  h^^y)  are  small 
compared  with  ai{ij)  and  a^iy)  and  may  be  neglected  in  the  right-hand 


A    LARGE    SIGNAL   THEORY   OP   TRAVELING-WAVE   AMPLIFIERS       355 


member  of  the  equation,  we  obtain  for  the  first  order  solution 


iKz,  t)  ^ 


ZqIo  I  (^ 


sin  <p  +  — ^^^  cos  <p 


40   \      2(1  +  bC)  I    dij  ^    '       dy 


(12b) 


Of  course,  the  solution  (12b)  is  justified  only  when  hi(y)  and  ?)2(?y)  thus 
obtained  are  small  compared  with  ai(y)  and  aoiy).  The  exact  solution 
of  B  obtained  by  successive  approximation  reads 


Biz,  t) 

+ 


ZqIo  I  c 


4(7  V     2(1  +  bC) 


4(1  +  hC) 
It  may  be  seen  that  the  term  involving 


dai(y)    .  ,   da2(ij) 

-^  sm  <p  +      ,       cos 
_    dy  dy 

■ ] 


•] 


'^^'  cos<p  +  — f^sm^     + 


(12c) 


dy-  dy- 


4(1  +  bcy 

and  the  higher  order  terms  are  neglected  in  our  approximate  solution. 
For  C  equal  to  few  tenths,  the  difference  between  (r2b)  and  (12c)  only 
amounts  to  few  per  cent.  We  thus  can  calculate  the  backward  wave  by 
(12b)  or  (12c)  from  the  derivatives  of  the  forward  wave.  To  obtain  the 
complete  solution  of  the  backward  wave,  we  should  add  to  (12b)  or 
(12c)  a  solution  of  the  homogeneous  equation.  We  shall  return  to  this 
point  later. 

4.  WORKING   EQUATIONS 

With  this  discussion  of  the  backward  wave,  we  are  now  in  a  position 
to  derive  the  working  equations  on  which  our  calculations  are  based.  In 
Nordsieck's  notation,  each  electron  is  identified  by  its  initial  phase. 
Thus,  (p(y,  (fo)  and  Cuow(y,  <po)  are  respectively  the  phase  and  the  ac 
velocity  of  the  electron  which  has  an  initial  phase  (fo  .  It  should  be  remem- 
bered that  y  is  equal  to 

and  is  used  by  Nordsieck  as  an  independent  variable  to  replace  the  vari- 
al)le  z.  Let  us  consider  an  electron  which  is  at  Zo  when  /,  =  0  and  is  at 
z  (or  ?/)  when  t  =  /.  Its  initial  phase  is  then 

OiZo 

<Po  =  — 


356 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


and  its  phase  at  y  is 


<p(y,<po)  =  oj  f-  -  tj 


i 


The  velocity  of  the  electron  is  expressed  as 

dz 


dt 


=  Wo[l  +  Cw{ij,  ip^)] 


where  Uo  is  the  average  velocity  of  the  electrons,  and,  Cuow(y,  tpo)  as  men- 
tioned earlier,  is  the  ac  velocity  of  the  electron  when  it  is  at  the  position 
y.  The  electron  charge  density  near  an  electron  which  has  an  initial  phase 
cpo  and  which  is  now  at  y,  can  be  computed  by  the  equation  of  conserva- 


tion of  charge,    it  is 


p(y,  <Po)  =  - 


Wo 


d(po 


d(p{y,  <po) 


1 


1  +  Cw(y,  ifo) 


(13) 


One  should  recall  here  that  h  is  the  dc  beam  current  and  has  been  de- 
fined as  a  positive  quantity.  When  several  electrons  with  different  initial 
phases  are  present  at  y  simultaneously,  a  summation  of 

d<po 


of  these  electrons  should  be  used  in  (13).  From  (13),  the  fundamental 
component  of  the  electron  charge  density  is 


pMt)  =  --- 


sm 


d<po 


sin  (fiy,  <po) 
1  +  Cw{y,  <pq) 

r^"   ,        cos  <p{y,  <po) 
+  cos  <p  I      d(po 
Jo 


(14) 


1  +  Cw(y,  ifo)/ 

These  are  important  relations  given  by  Nordsieck  and  should  be  kept 
in  mind  in  connection  with  later  work.  In  addition,  we  shall  frequently 
use  the  transformation 

I  =  t  s  =  ^"(' +  ^'"(^-»  1^ 

which  is  written  following  the  motion  of  the  electron.  Let  us  start  from 
the  forward  wave.  It  is  computed  by  means  of  (6).  After  substituting 
(8)  and  (14)  into  (6),  we  obtain  by  equating  the  sin  <p  and  the  cos  v' 


A   LARGE   SIGNAL   THEORY   OF   TRAVELING-WAVE   AMPLIFIERS       357 

terms 

dax{y)  ^  _2    T^"  ^        sin  <p(y,  cpo)  .^. 

dy  IT  h  "  1  +  Cwiv,  (po) 

da.Xy)  2   f^"  cos<p{y,<po)  ..^n 

— 1 —  =   ~-  /      d(po  , r  (.lb; 

dy  IT  Jo  1  +  Cw{y,  <po) 

Next  we  shall  calculate  the  electron  motion.  We  express  the  acceleration 
of  an  electron  in  the  form 

d'z        „      /I    I    /o    /        ^^  dw{y,  <po) 
^  =  Cuod  +  Cw{y,  M  -^^ 

and  calculate  the  circuit  field  by  differentiating  F  in  (8)  and  B  in  (12c) 
with  respect  to  z.  One  thus  obtains  from  Newton's  law 

2[1  +  Cw{y,  <po)]  ^^'^J'  ^°^  =  (1  +  hOMy)  sin  <p  +  a,{y)  cos  <p\ 

dy 

+  ^-^  r^  «in  ^  +  ^^  cos  J  -  -^  ^. 
4(1  +  6C)  L    ^Z/-  c?^^  J       WomwC^ 

Here  Eg  is  the  space  charge  field,  which  will  be  discussed  in  detail  later. 
Finally  a  relation  between  w{y,  (po)  and  <p{y,  ^o)  is  obtained  by  means  of 
(13) 

difiy,  <po)  _  ^  ^        ^^(y,  <Po)  QgN 

dy  1  +  Cw{y,  <pq) 

Equations  (15),  (16),  (17)  and  (18)  are  the  four  working  equations 
which  we  have  derived  for  finite  C  and  including  space  charge. 

Instead  of  writing  the  equations  in  the  above  form,  Rowe,  ignoring 
the  backward  wave,  derives  (15)  and  (16)  directly  from  the  circuit 
equation  (1).  He  obtains  an  additional  term 

C  d^tti 


2  dy"" 


for  (15)  and  another  term 


C  d"ai 

2df 

for  (16).  It  is  apparent  that  the  backward  wave,  though  generally  a 
small  quantity,  does  influence  the  terms  involving  C. 


358 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


5.   THE   SPACE   CHARGE   EXPRESSION 


We  have  mentioned  earlier  that  the  space  charge  field  is  computed 
from  the  disc-model  suggested  by  Tien,  Walker  and  Wolontis.  In  their 
calculation,  the  force  excited  on  one  disc  by  the  other  is  approximated 
by  an  exponential  function 


F.  = 


—  [a(z'— z)/ro] 


27rro-eo 


Here  ro  is  the  radius  of  the  disc  or  the  beam,  q  is  the  charge  carried  by 
each  disc,  and  eo  is  the  dielectric  constant  of  the  medium.  The  discs  are 
supposed  to  be  respectively  at  z  and  z  .  a  is  a  constant  and  is  taken 
equal  to  2. 

Consider  two  electrons  which  have  their  initial  phases  <pq  and  ^o  and 
which  reach  the  position  ij  (or  z)  at  times  t  and  t'  respectively.  The  time 
difference, 


*  -  /  =  1 

00 


wt    —    —  Z   —   [bit     —   —  Z) 

Vo  \  Vq        J 


CO 


multiplied  by  the  velocity  of  the  electron  i<o[l  +  Cw(y,  (po  )]  is  obviously 
the  distance  between  the  two  electrons  at  the  time  t.  Thus 

(z    -  z)t=t  =  -  y(y,  <Po)  -  <p(y,  <Po)]uo[l  +  Cw(y,ipo)]       (19a) 

In  this  equation,  we  are  actually  taking  the  first  term  of  the  Taylor's 
expansion, 


(z    —  z)t=t  = 


dzjij,  cpo) 
dt 


t=t 


(t  _  /^  j_  ^  c?^2(y,  <pq) 


it  -  ty 


t=t 


(19b) 


+ 


It  is  clear  that  the  electrons  at  y  may  have  widely  different  velocities 
after  having  traveled  a  long  distance  from  the  input  end,  but  changes  in 
their  velocities,  in  the  vicinity  of  y  and  in  a  time-period  of  around  2  tt, 
are  relatively  small.  This  is  why  we  must  keep  the  first  term  of  (19b) 
and  may  neglect  the  higher  order  terms.  From  (19a)  the  space  charge 
field  Es  in  (17)  is 


2e 


Es  = 


/+00 


-k]ip(.y  ,<po+<t>)—<p(.U  ,<Po)  1  li+Cw(y,(po+<t>)] 


d(f>  sgn  (<p(<po  -\-  <p)  -  <t>iy,  <po)) 
Here,  e/m  is  the  ratio  of  electron  charge  to  mass,  cop  is  the  electron 


A    LARGE    SIGNAL   THEORY    OF   TRAVELING-WAVE    AMPLIFIERS       359 

angular  plasma  frequency  for  a  beam  of  infinite  extent,  and  k  is 

2 


k  = 


a 

0)  CO 

—  ro  —  ro 

Uo  Wo 


(20) 


In  the  small  C  theory,  th^e  distribution  of  electrons  in  time  or  in  time- 
phase  at  z  is  approximately  the  same  as  the  distribution  in  z  (also  ex- 
pressed in  the  unit  of  time-phase)  at  the  vicinity  of  z.  This  is,  however, 
not  true  when  C  becomes  finite.  The  difference  between  the  time  and 
space  distributions  is  the  difference  between  unity  and  the  factor 
(1  -}-  Cw{y,  <po )).  We  can  show  later  that  the  error  involved  in  con- 
;  sidering  the  time  phase  as  the  space  phase  can  easily  reach  50  per  cent 
or  more,  depending  on  the  velocity  spread  of  the  electrons. 


6.  NUMERICAL   CALCULATIONS 

Although  the  process  of  carrying  out  numerical  computations  has 
been  discussed  in  Nordsieck's  paper,  it  is  desirable  to  recapitulate  here 
I  a  few  essential  points  including  the  new  feature  added.  Using  the  work- 
ing equations  (15),  (16),  (17)  and  (18), 

dai    da 2    dw  ,         dcp 

dy  '  dy  '  dy  dy 

\  are  calculable  from  ai ,  a^ ,  w  and  <p.  The  distance  is  divided  into  equal 
I  intervals  of  A?/,  and  the  forward  integrations  of  Oi ,  ao  ,  w  and  (p  are  per- 
f  formed  by  a  central  difference  formula 


ax{y  +  A?/)  =  ax{y)  -f 


dy 


y+y2&y 


■Ay 


In  addition. 


d^ai 
dy^ 


and 


d  02 

df 


in  (17)  are  computed  from  the  second  difference  formula  such  that 
d''ai 


-  At/ 


_     dtti  da\ 

dy^      j/=j/  \_dy       y+l/2i,y  dy      y-^/2^y_ 

We  thus  calculate  the  behavior  along  the  tube  by  forward  integration 
j  made  in  steps  of  Ay,  starting  from  y  =  0.  At  ?/  =  0  the  initial  condi- 
tions are  determined  from  Pierce's  linearized  theory.  Because  of  its 
complications  in  notation,  this  will  be  discussed  in  detail  in  Appendix  I. 
j     Numerical  calculations  were  carried  out  using  the  701-type  I.B.M. 


Table  I 


a; 
U 

QC 

k 

c 

6 

Ml 

MJ 

ycsAT.) 

<! 
01! 

H 
•i 

i 

a. 
1 

1 

0.1 

2.5 

0.05 

0.455 

m  max. 
0.795662 

-0.748052 

5.6 

1.26 

0.415 

2 

0.1 

2.5 

0.1 

0.541 

Ml  max. 
0.827175 

-0.787624 

5.2 

1.24 

0.482 

3 

0.1 

2.5 

0.1 

1.145 

0.941;ui  max. 
0.778535 

-1.05370 

5.6 

1.31 

0.820 

4 

0.1 

2.5 

0.1 

1.851 

0.66jui  max. 
0.550736 

-1.37968 

7.0 

1.36 

1.05 

J 

5 

0.1 

2.5 

0.2 

0.720 

m  max. 
0.900312 

-0.873606 

4.8 

1.02 

0.726 

6 

0.2 

1.25 

0.1 

0.875 

jui  max. 
0.769795 

-1.04078 

5.9 

1.22 

0.570 

7 

0.2 

1.25 

0.1 

1.422 

0.951^1  max. 
0.724527 

-1.29469 

6.0 

1.30 

0.803 

8 

0.2 

1.25 

0.1 

2.072 

0.666mi  max. 
0.512528 

-1.60435 

7.6 

1.35 

1.08 

9 

0.2 

2.5 

0.05 

0.765 

Ml  max. 
0.731493 

-0.973376 

6.2 

1.30 

0.412 

10 

0.2 

2.5 

0.1 

0.875 

Ml  max. 
0.769795 

-1.04078 

5.8 

1.22 

0.490 

11 

0.2 

2.5 

0.1 

1.422 

0.941mi  max. 
0.724527 

-1.29469 

6.0 

1.26 

0.720 

12 

0.2 

2.5 

0.1 

2.072 

0.666mi  max. 
0.512528 

-1.60435 

7.2 

1.25 

0.92 

13 

0.2 

2.5 

0.1 

2.401 

0.300mi  max. 
0.230930 

-1.76243 

12.4 

1.24 

1.36 

j 

U 

0.2 

2.5 

0.15 

0.976 

Ml  max. 
0.812900 

-1.10656 

5.4 

1.11 

0.572 

15 

0.2 

2.5 

0.15 

1.549 

0.941mi  max. 
0.765101 

-1.37540 

5.8 

1.14 

1.03 

16 

0.2 

2.5 

0.15 

2.2311 

0.666mi  max. 
0.541234 

-1.70180 

7.0 

1.12 

1.22 

17 

0.2 

2.5 

0.15 

2.575 

0.300mi  max. 
0.243864 

-1.86844 

10.8 

1.04 

1.34 

18 

0.4 

2.5 

0.05 

1.25 

Ml  max. 
0.653014 

-1.36746 

7.6 

1.26 

0.315 

19 

0.4 

2.5 

0.1 

1.38 

Ml  max. 
0.701470 

-1.47477 

6.6 

1.11 

0.674 

20 

0.4 

2.5 

0.1 

1.874 

0.941mi  max. 
0.660223 

-1.71341 

7.8 

1.19 

1.05 

21 

0.4 

2.5 

0.1 

2.458 

0.666mi  max. 
0.467038 

-1.99840 

8.6 

1.09 

1.25 

l> 


360 


A    LARGE    SIGNAL   THEORY    OF   TRAVELING- WAVE   AMPLIFIERS       361 

equipment.  The  problem  was  programmed  by  Miss  D.  C.  Legaus.  The 
cases  computed  are  listed  in  Table  I  in  which  m  and  m2  are  respectively 
Pierce's  .xi  and  iji ,  and  A,(d  —  iny)  and  tj  at  saturation  will  be  discussed 
later.  All  the  cases  were  computed  with  A^  =  0.2  using  a  model  based 
on  24  electron  discs  per  electronic  wavelength.  To  estimate  the  error 
involved  in  the  numerical  work,  Case  (10)  has  been  repeated  for  48  elec- 
trons and  Cases  (10)  and  (19)  for  Ay  =  0.1.  The  results  obtained  by 
using  different  numbers  of  electrons  are  almost  identical  and  those  ob- 
tained by  varying  the  inter\'al  A//  indicate  a  difference  in  A  (y)  less  than 
1  per  cent  for  Case  (10)  and  about  6  per  cent  for  Case  (19).  As  error 
generally  increases  with  QC  and  C  the  cases  listed  in  this  paper  are 
limited  to  QC  =  0.4  and  C  =  0.15.  For  larger  QC  or  C,  a  model  of  more 
electrons  or  a  smaller  interval  of  integration,  or  both  should  be  used. 

7.    POWER   OUTPUT   AND    EFFICIENCY 

Define 

A(ij)  =  HVa,(yy  +  aM' 

-0(y)=i^n-'^-^  +  by  ^^^^ 

aiiy) 


We  have  then 


F{z,t)  =  ^A{y)  cos 


^  -^t-  e{y) 

Uo 


(22) 


The  power  carried  by  the  forward  wave  is  therefore 

2CA'hVo  (23) 


(f)      = 

\Z/o/  average 


and  the  efficiency  is 


Eff.  =  ?£^^  =  2CA'        or        ^  =  2CA'  (24) 

In  Table  I,  the  values  of  A(y),  6{y)  and  y  at  the  saturation  level  are 
listed  for  every  case  computed.  We  mean  by  the  saturation  level,  the 
distance  along  the  tube  or  the  value  of  y  at  which  the  voltage  of  the 
forward  wave  or  the  forward  traveling  power  reaches  its  first  peak. 
The  Eff./C  at  the  saturation  level  is  plotted  in  Fig.  1  versus  QC,  for 
k  =  2.5,  h  for  maximum  small-signal  gain  and  C  =  small,  0.05,  0.1,  0.15 
and  2.  It  is  also  plotted  versus  h  in  Fig.  2  for  QC  =  0.2,  k  =  2.5  and 
C  =  small,  0.1  and0.15,  and  in  Fig.  3  for  QC  =  0.2,  C  =  0.1  and  k  =  1.25 
and  2.50.  In  Fig.  2  the  dotted  curves  indicate  the  values  of  h  at  Avhich 


1 


362  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    195G 

4.5 


0.5 


Fig.  1  —  The  saturation  eff./C  versus  QC,  for  k  =  2.5,  h  for  maximum  small- 
signal  gain  and  C  =  small,  0.1,  0.15  and  0.2. 

ixx  =  Ml  (max),  0.94  jui(max),  0.67  iui(max)  and  0.3  /ii(niax),  respectively. 
It  is  seen  that  Eff./C  decreases  as  C  increases  particularly  when  h  is 
large.  It  is  almost  constant  between  k  =  1.25  and  2.50  and  decreases 
slowly  for  large  values  of  C  when  QC  increases. 

The  (Eff./C)  at  saturation  is  also  plotted  versus  QC  in  Fig.  4(a)  for 
small  C,  and  in  Fig.  4(b)  for  C  =  0.1.  It  should  be  noted  that  for  C  =  0.1 
the  values  of  Eff./C  fall  inside  a  very  narrow  region  say  between  2.5  to 
3.5,  whereas  for  small  C  they  vary  widely. 

8,  VELOCITY   SPREAD 

In  a  traveling-wave  amplifier,  when  electrons  are  decelerated  by  the 
circuit  field,  they  contribute  power  to  the  circuit,  and  when  electrons 
are  accelerated,  they  gain  kinetic  energy  at  the  expense  of  the  circuit 
power.  It  is  therefore  of  interest  to  plot  the  actual  velocities  of  the  fastest 
and  the  slowest  electrons  at  the  saturation  level  and  find  how  they  vary 
with  the  parameters  QC,  C,  b  and  k.  This  is  done  in  Fig.  5.  These  veloci- 
ties are  also  plotted  versus  y  for  Case  10  in  Fig.  6,  in  which,  the  A(y) 
curve  is  added  for  reference. 

9.  THE  BACKWARD  WAVE  AND  THE  FUNDAMENTAL  COMPONENT  OF  THE 
ELECTRON  CHARGE  DENSITY 

Our  calculation  of  efficiency  has  been  based  on  the  power  carried  by 
the  forward  wave  only.  One  may,  however,  ask  about  the  actual  power 


A    LARGE   SIGNAL   THEORY   OF   TRAVELING-WAVE   AMPLIFIERS       363 


6.0 
5.5 

5.0 
4.5 
4.0 

3.5 

EFFI. 

C       3.0 
(SAT.) 

2.5 
2.0 
1  .5 
1.0 
0.5 


1 

QC  =  0.2 

1 

1 

A- — 

K=2.5 

\ 

SMALI " 

*^r^ 

\ 
1 

Sa 

y^ 

1 

1 

\ 

_/^ 

I 

Ji 

A 

1 

/^ 

\ 

1 
\ 

/ 

\ 

t 

\ 

/ 

\ 

\ 

^ 

/ 

\ 

\ 

X 

f 

\ 
( 
\ 

\ 
\ 

\ 

^   ' 

\ 

\ 

C  =  0.1 

'\ 

\ 

\ 

\ 

, 

lyj 

\ 

\ 

\ 

JT"^ 

\ 

\ 

V    C=0.15 

\ 
\ 
\ 

\ 

\ 
\ 

>"1  = 

1 
AX) 

/"1=C 

K      1             1 
).94/Z.(MAX) 

.at^ 

/     1 

\^ 

/t/i  =  0.67//i(MAX) 

\ 

//,  =  0.3//i(MAX) 

0.5 


1.0 


1.5 

b 


2.0 


2.5 


3.0 


Fig.  2  —  The  saturation  eff./C  versus  fe,  for  k  =  2.5,  QC  =  0.2,  and  C  =  small, 
0.1  and  0.15.  The  dotted  curves  indicate  the  values  of  h  for  m  =  \,  0.94,  0.67,  and 
0.3  of  ;ui(max)  respectively. 

output  in  the  presence  of  the  backward  wave.  For  simphcity,  we  shall 
use  the  approximate  solution  (12b)  which  can  be  written  in  the  form 

B{z,  t)  ^  Real  Component  of 

ZqIq        c 


4C  2(1  +  hC) 


dax(y)Y  ^  (da,{y)\-  j^^-v,.-,y+j^\     (12d) 


with 


tan  ^  = 


dij 


(laiiyT 
dy    , 


dy 


dchiyY 
dy    , 


As  mentioned  earlier  that  the  complete  solution  of  (6)  is  obtained  by 
adding  to  (12b)  a  complementary  function  such  that 


-yu  1+  r  Qz 


ZqIq 


+ 


c 


4C  2(1  +  bC) 


dy:)  ^\dy  )  ' 


-hy+ji 


(25) 


364  THE   BELL    SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


EFFI. 

c 

(SAT.)  3 


QC  =  o.2 
C  =  0.1 

J<_=K25. 

3- 

2.50 

0.6       0.8       1.0         1.2        1.4         1.6         1.8       2.0       2.2       2.4 

b 

Fig.  3  —  The  saturation  eff./C  versus  b,  for  QC  =  0.2  C  =  0.1  and  k  =  1.25 
and  2.50. 

If  the  output  circuit  is  matched  by  cold  measurements,  the  backward 
wave  must  be  zero  at  the  output  end,  z  =  D.  This  determines  Ci ,  that  is, 


„  ZqIq         c 

^1  =  ~^rPT 


or 


Cie 


jut+Toz 


4C  2(1  +  bC) 


ZqIq  C 


//dai(t/)Y        I    /da2{y)Y        ro(2+bc)D+ji 


dai{y)V  /da2{y)y 


4C  2(1  +  6C)    y   \    dy    )z=o       \    dy    Jz^d  (26) 

The  backward  wave  therefore  consists  of  two  components.  One  compo- 


o 

7 

(a) 

C  =  SMALL 

^- 

Ml  =  0.67 

/U,(MAX) 

D 

5 

^^;;:^ 

^^ 

EFFI. 
C       4 

/U,  =  0.94//i(MAX) 

^^^ 

1 



(SAT.)  ^ 
2 

■"Zr^AtlC^AX) 

" 

0 

(b) 

C  =  o.i 

=  0.94//,  (MAX) 

1 

Xj 

fea,^_^-VZi  =  0.67 /Z,  (MAX) 

>U,  =  /i|(MAX)- 

1       —===3 

^^^ 

0.1 


0.2 

QC 


0.3  0.4     0  0.1 


0.2 

QC 


0.3  0.4 


Fig.  4  —  The  saturation  eff./C  versus  QC  for  h  corresponding  jui  =  1,  0.94  and 
0.67  of  Mi(max),  (a)  for  C  =  small,  (b)  for  C  =  0.1. 


A   LARGE   SIGNAL   THEORY   OF   TRAVELING- WAVE   AMPLIFIERS       365 


nent  is  coupled  to  the  beam  and  has  an  amplitude  equal  to 

Zolo         C 
IC  2(1  +  bC) 


VX^'Y  + 


K^y  / 


\dy) 


which  generally  grows  with  the  forward  wave.  It  thus  has  a  much  larger 
amplitude  at  the  output  end  than  at  the  input  end.  The  other  component 
is  a  wave  of  constant  amplitude,  which  travels  in  the  direction  opposite 
to  the  electron  flow  with  a  phase  velocity  equal  to  that  of  the  cold  cir- 
cuit. At  the  output  end,  2  =  Z),  both  components  have  the  same  ampli- 
tude but  are  opposite  in  sign.  One  thus  realizes  that  there  exists  a  re- 
flected wave  of  noticeable  amplitude,  in  the  form  of  (26),  even  though 
the  output  circuit  is  properly  matched  by  cold  measurements.  Under 
j  such  circumstances,  the  voltage  at  the  output  end  is  the  voltage  of  the 
forward  wave  and  the  power  output  is  the  power  carried  by  the  forward 
wave  only.  This  is  computed  in  (23). 
Since  (26)  is  a  cold  circuit  wave  it  may  be  eliminated  by  properly  ad- 


c[-w], 


■C[w], 


5.0 


4.5 


4.0 


3.5 


5  3.0 

2 

o 

9-  2.5 


1.5 


1.0 


0.5 


(a) 

; 

/ 

y 

/ 

/ 

( 

r' 

,.--- 

( 

L"1 

.-'■ 

(b) 

j^ 

V 

/ 

/ 

/ 

y 

1 

Qw 

'"--^ 

^-"^ 

(c) 

J 

i 

/ 

/ 

^ 

/ 

/ 

,''^ 

1 

/        ( 

f 

r 

1 
1 
1 

1 

< 

f 

0.1  0.2         0.3       0.4    0.5         1.0 

QC 


1.5         2.0         2.5    0 
b 


0.05      0.10         0.15        0.20 


Fig.  5  —  Cw(y,  <po)  of  the  fast  and  the  slowest  electrons  at  the  saturation  level, 
(a)  versus  QC  for  k  =  2.5,  C  =  0.1  and  b  for  maximum  small-signal  gain;  (b)  versus 
6  for  A;  =  2.50,  C  =  0.1  and  QC  =  0.2;  and  (c)  versus  C  for  A-  =  2.50,  QC  =  0.2 
and  b  for  maximum  small-signal  gain. 


366  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1950 


3.5 


3.0 


2.5 


2.0 


9- 

^1.5 
U 

1.0 


0.5 


> 

r\ 

/ 

^ 

\ 

/ 

\ 

CASE  10 
QC  =  0.2 
C  =  0.1 
b  =  0.875 
k  =  2.5 

MAXC(-W)  / 

,-' 

''s 

\ 

\ 

/A(y) 

\ 
\ 

, 

1 

1 

/ 

/\ 

S 

// 

y 

/ 

X- 

./ 

,^ 

"^AXCW 

i 
/ 

/ 

/ 

y 

r 

■7 

/ 

/ 

A 

y 

^-' 

^ 

^ 

:z=^ 

—  **       — 

^ 

1.4 


1.2 


1.0 


0.8 


ID 
< 


0.6 


0.4 


0.2 


0 
0  0.5        1.0  1.5        2.0        2.5        3.0        3.5       4.0        4.5        5.0        5.5        6.0        6.5        7.0         7.5 

y 

Fig.  6  —  Cw{y,  (pa)  of  the  fast  and  the  slowest  electrons  versus  y  for  Case 
(10).  A{y)  is  also  plotted  in  dotted  lines  for  reference. 

justing  the  impedance  of  the  output  circuit.  This  may  be  necessary  in 
practice  for  the  purpose  of  avoiding  possible  regenerative  oscillation.  In 
doing  so,  the  voltage  at  2  =  D  is  the  sum  of  the  voltage  of  the  forward 
wave  and  that  of  the  particular  solution  of  the  backward  wave.  In  every 
case,  the  output  power  is  always  equal  to  the  square  of  the  net  voltage 
actually  at  the  output  end  divided  by  the  impedance  of  the  output  cir- 
cuit. 

We  find  from  (14),  (15)  and  (16)  that  the  fundamental  component  of 
electron  charge  density  may  be  written  as 

f     s.        \  h  (  .       dai{y)    .  da2(y)\ 


=  Real  component  of 


1/0 


dai{y) 
dy    , 


+ 


doM 
dy 


(26) 


jo)—Toz—by+Ji 


) 


where  —Io/uq  is  the  dc  electron  charge  density,  po  . 

If  (26)  is  compared  with  (12d)  or  (12c),  it  might  seem  surprising  that 
the  particular  solution  of  the  backward  wave  is  just  equal  to  the  funda- 


A   LARGE   SIGNAL   THEORY   OF   TRAVELING-WAVE   AMPLIFIERS       367 


1.6 
t.5 
1.4 
1.3 
1.2 
1.1 
1.0 

Pq  0.8 
0.7 
0.6 
0.5 
0.4 
0.3 
0.2 
0.1 


1.2 
1.1 
1.0 
0.9 

Pq  0.7 
0.6 
0.5 
0.4 
0.3 
0.2 

0.1 

0 


CASES  2, 

10,19 

(a) 

k  =  2.5 
-       C  =  0.1 
b-»MAX  n^ 

/' 

\ 

r 

\ 

J 

' 

^ 

r 

\ 

r\ 

\ 

/ 

f 

V 

1] 

s — 

QC=o.i/ 

/ 

f 

0.2 

// 

r 

0.4 

// 

1 

V 

// 

\ 

7 

\ 

<^ 

L 

/ 

\ 

A 

\ 

^ 

^ 

CASES  9, 

0,14 

(c) 

QC=0.2 
-           k=2.5 
b-»MAX//| 

rv 

-\ 

1 

u 

\ 

r 

C=o,s/// 

\ 

/  rf-o  10 

\ 

///o.05 

i 

II 

k 

A 

f 

/ 

^ 

8    0 


4 

y 


CASES  1C 

,11,12 

iH 

r 

V 

r\ 

(b) 

QC  =  o.2 
C  =  o.i 

k  =  2.5 

r 

k\ 

(A 

\ 

// 

\  / 

y 

^ 

c 

\ 

/ 

\f 

A 

\ 

\ 

/^,  =  >U,MAx/^ 

' 

/ 

A 

/ 

\ 

// 

/ 

\ 

/ 

f 

\ 

\ 

A 

/ 

I 

/ 

J 

/09«   / 

11 

y 

/ 

\ 

\    . 

// 

/ 

/ 

11 

/ 

\J 

17 

// 

Ai.^i 

^^ 

"w 

1 

/' 

y 



^^ 

.^^ 

-^ 

>^ 

'^1  =  0.3X/,MAX 
1             1 

7  8 

y 


10         11 


12        13        14         15 


Fig.  7(a)  — p^/po  versus  ?/,  (a)  using  QC  as  the  parameter,  for  A;  =  2.5,  C  =  0.1, 
and  6  for  maximum  small-signal  gain  (Cases  2, 10,  and  19) ;  (b)  using  h  as  the  param- 
eter, for  k  =  2.50,  C  =  0.1  and  QC  =  0.2  (Cases  10,  11,  12  and  13);  and  (c)  using 
C  as  the  parameter,  for  k  =  2.50,  QC  =  0.2  and  h  for  maximum  small-signal  gain 
(Cases  9,  10  and  14). 


368 


THE   BELL   SYSTEM   TECHNICAL  JOURNAL,   MARCH    1956 


mental  component  of  the  electron  charge  density  of  the  beam  multiplied 
by  a  constant 

/     Zq/o  C         2uo 


2wo\ 
h) 


(27) 


V     4C  2(1  +  hC) 
The  ratio  of  the  electron  charge  density  to  the  average  charge  density, 

P«(2) 


Po 


2319^21 
5  17/^,1  9 


^  +e 


Fig.  8(a)  —  y  versus  <f  -  hrj  for  QC  =  0.2,  k  =  2.5,  b  for  mi  =  0.67 
C  =  small. 


Ml  (max)  and 


A   LARGE   SIGNAL   THEORY   OF   TRAVELING-WAVE   AMPLIFIERS       369 

is  plotted  in  Fig.  7  versus  y,  using  QC,  h  and  C,  as  the  parameters.  They 
lare  also  the  curves  for  the  backward  wave  (the  component  which  is 
!  coupled  to  the  beam)  when  multiplied  by  the  proportional  constant  given 
in  (27).  It  is  interesting  to  see  that  the  maximum  values  of  p^/po  are 
between  1.0  and  1.2  for  QC  =  0.2  and  decrease  as  QC  increases.  The 
peaks  of  the  curves  do  not  occur  at  the  saturation  values  of  y. 

10.  y  VERSUS  ((p  —  by)  diagrams 

To  study  the  effect  of  C,  b,  and  QC  on  efficiency  y  versus  (<p  —  by) 
diagrams  are  plotted  in  Figs.  8(b),  (c),  (d)  and  (e)  for  Cases  (21),  (16), 
(10)  and  (21),  respectively.  {<p  —  by)  here  is  ($  +  6)  in  Nordsieck's  nota- 
tion. In  these  diagrams,  the  curves  numbered  from  1  to  24  correspond  to 
the  24  electrons  used  in  the  calculation  with  each  curve  for  one  electron. 
Only  odd  numbered  electrons  are  presented  to  avoid  possible  confusion 
arisen  from  too  many  lines.  The  reciprocal  of  the  slope  of  the  curve  as 


-10      -9      -8 


jo-by 


Fig.  8(b)  —  y  versus  <p 
C  =  0.1  (Case  12). 


bij  for  QC  =  0.2,  k  =  2.5,  b  for  mi  =  0.67Mi(max)  and 


370 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    1956 


given  by  (18)  is  proportional  to  the  ac  displacement  of  electron  per  unit 
of  ij.  (In  small-C  theorj^  it  is  proportional  to  the  ac  velocity  of  the  elec- 
tron.) Concentration  of  curves  is  obviously  proportional  to  the  charge- 
density  distribution  of  the  beam.  In  the  shaded  regions,  the  axially  di- 
rected electric  field  of  the  circuit  is  negative  and  thus  accelerates  elec- 
trons in  the  positive  z  direction.  Electrons  are  decelerated  in  the  un- 
shaded regions  where  the  circuit  field  is  positive.  The  boundaries  of  these 
regions  are  constant  phase  contours  of  the  circuit  wave.  (They  are  con- 
stant $  contours  in  Nordsieck's  notation.) 

These  figures  are  actuallj'  the  "space-time"  diagrams  which  unfold 
the  historj^  of  every  electron  from  the  input  to  the  output  ends.  The 
effect  of  C  can  be  clearly  seen  by  comparing  Figs.  8(a),  (b)  and  (c). 
These  diagrams  are  plotted  for  QC  =  0.2,  A;  =  2.5,  h  for  jui  =  0.67 
jui(max)  and  for  Fig.  8(a),  C  =  small,  for  Fig.  8(b),  C  =  0.1,  and  for 
Fig.  8(c),  C  =  .15.  It  may  be  seen  that  because  of  the  velocity  spread  of 
the  electrons,  the  saturation  level  in  Fig.  8(a)  is  9.3  whereas  in  Figs.  8(b) 
and  (c),  it  is  7.2  and  7.0,  respectively.  It  is  therefore  not  surprising  that 
Eff./C  decreases  as  C  increases. 

The  effects  of  h  and  QC  may  be  observed  by  comparing  Figs.  8(d)  and 
(b),  and  Figs.  8(b)  and  (e),  respectively.  The  details  will  not  be  de- 
scribed here.  It  is  however  suggested  to  study  these  diagrams  with  those 
given  in  the  small-C  theory. 


7.2 

5 

1 

23  9 

11 

i'5 

7     3 

1719  21  13 

23 

15 

1719  21 

^" 

^-^^  ?ny 

J^ 

V 

^v:\ 

S 

\| 

A 

\- 

I 

SATURATION 

6.8 

6.4 
6.0 
5.6 
6.2 

.«,^ 

^ 

*tf 

LEVEL 

" 

vK 

sL- 

^ 

^N 
^ 

V 

\ 

^ 

■I 

^ 

^ 

L^ 

i 

^ 

^ 

y 

r 

\ 

rt 

'a 

[  \ 

1 

rt 

^VL 

/ 

1 

V 

«x 

t 

/ 

< 
$ 

^W 

I       / 

/ 

-^ 

\ 
\ 

\ 

w 

t 

ll  1 

'—  T 

^ 

kU\ 

4.4 

4.0 

3.6 

3.2 

2.8 

2.4 
?0 

'^ 

t 

^ 

\\\^ 

\  \   V 

\ 

I  1 

/ 

\\ 

1 
-    1 

\  \ 
1  \ 
1  \ 

r- 

1 

/, 

1 

\ 

\\\ 

IS     _H 

1  1 

li 

V-' 

\  \ 

\ 

' 

% 

1 

■  1 

1    1 
1    1 

i     1 

>  1 
1 

\ 

\ 

1 

j 

i 

r 
1 

ll 

i  i 

i5!r 

1 

ti23l 
11     1 

,3 

_ 

-1 

9  in 

L.          J 

3  15  17  ig/sii 
1         i     i 

23 

-10     -9 


-8      -7 


-4     -3 


0        1 


10 


Fig.  8(c)  —  y  versus  <p  —  by  for  QC  =  0.2,  k  —  2.5,  b  for^i  =  0.67^1  (max)  and  , 
C  =  0.15  (Case  16). 


A   LARGE   SIGNAL   THEORY    OF   TRAVELING-WAVE   AMPLIFIERS       371 
11.    A    QUALITATIVE    PICTURE    AND    CONCULSIONS 

We  have  exhibited  in  the  previous  sections  the  most  important  non- 
linear characteristics  of  the  traveling  wave  ampUfier.  Xumerical  compu- 
tations based  on  a  model  of  24  electrons  have  been  carried  out  for  more 
than  twenty  cases  covering  useful  ranges  of  design  and  operating  parame- 
ters. The  results  obtained  for  the  saturation  Eff./C  may  be  summarized 
as  follows: 

(1)  It  decreases  with  C  particularly  at  large  values  of  QC. 

(2)  For  C  =  0.1,  it  varies  roughly  from  3.7  for  QC  =  0.1  to  2.3  for 
i}C  =  0.4,  and  only  varies  slightl}^  with  h. 

(3)  For  C  =  0.15,  it  varies  from  2.7  to  2.5  for  QC  from  0.1  to  0.2  and 
\i  corresponding  to  the  maximum  small-signal  gain.  It  varies  slightly 
with  h  for  QC  =  0.2. 

(4)  It  is  almost  constant  between  k  —  1.25  and  2.50. 

In  order  to  understand  the  traveling-wave  tube  better,  it  is  important 
to  have  a  simplified  qualitative  picture  of  its  operation.  It  is  obvious  that 
to  obtain  higher  amplification,  more  electrons  must  travel  in  the  region 
where  the  circuit  field  is  positive,  that  is,  in  the  region  where  electrons 


6.8 
6.4 
6.0 

17 

3 
51    9i7 

13     15 

11 

23     21 

7 
J 19 

13 

5 

11 

O^ 

N 

N 

cl. 

\ 

vV 

\ 

vn 

^ 

.     ^^vV    ^ 

\ 

Vv 

\ 

\ 

^A 

TURAT 

lOM 

^ 

v^§^ 

\ 

\ 

\j 

l\ 

LEVEL 

5.6 
5.2 
4.8 
4.4 
4.0 
3.6 
3.2 
2.8 
2.4 

* 

3"  - 

. 

/ 

^ 

\ 

\ 

N 

/ 

/ 

/ 

/ 

'V 

\\ 

\ 

K 

l\ 

\ 

1 
1 

1 

1 

/ 

/ 
1 

/i^ 

\\ 

-^A 

"^ 

\ 

\ 
t 

/ 
/ 
1 

r 
1 

■  /A\\\ 

f 

f 

fX 

V 

v 

1 
1 

\ 

1 
I 
1 

/ 

N 

\\v 

\\  \ 

/ 

t 

/ 

1 

\ 

^ 

\°  1 

I 

i 

/ 
f 

\\\ 

f 

1 

/ 

\  \ 

\  \ 

f 

1 
1 

( 

1 

r 

f 

// 

\\ 

1 
1 

1 

\\ 

/ 

\  \ 

11«13» 

151    17 

1      3 

5 

] 

g\  inisl  15 

7 

19 

21/  2 

3 

?n 

1      1 

1 

1     \      li\ 

1    i 

t    J    iJ 

-1 


SP-by 


Fig.  8(d)  —  V  versus  <p  —  by  for  QC  =  0.2,  k  =  2.5,  b  for  m  =  mi  (max)  and 
''  =  0.1  (Case  10). 


372 


THE    BELL    SYSTEM   TECHNICAL    JOURNAL,    MARCH    1956 


8.8 

11 

5   3  15  9 

7  21 

17? 

23 

1 

15 

21 

23 

^-~ 

-^ 

R 

StiC^  1 

^^ 

/ 

/' 

1- 

SATURATION 

8.4 
8.0 
7.6 
7.2 
6.8 
6.4 
6.0 
6.6 
5.2 
4.8 
4.4 
4.0 
3.6 
3.2 
2.8 
2.4 
?,0 

IQ** 

:^ 

^ 

■"           LEVEL 

~    - 

1 

r4- 

■~3 

N,    , 

/ 

\ 

\ 

\ 

\, 

7H 
1 

"^^ 

d 

\ 

/ 

^v.-— ^.-. 

""^ 

\ 
\ 

t 

y- 

[V\ 

fe 

\ 

^ 

\ 

>^ 

\ — p 

\  \ 
\  \ 

v\ 

I 

:: 

''^: 

N\ 

"t 

^ 

:\ 

\ 
\ 

-^^ 

K\l 

1 

H. 

'■.-■; 

1 

1 
1 

Ui 

y 

V 

t 
1 
J. 

1  ^W 

^\ 

/ 

/ 

)) 

r 
i 
t 

l-^i 

\V 

d 

\ 

r 

// 

1 
( 
1 

i  : 

1 

\\ 

; 

1 

/ 

i 

w 

\\ 

,  :' 

■,J; 

W 

I 

1 
1 
1 

— p- 

rl  - 

1    \ 

/ 

] 

\ 

-— - 

1 

1 
1 

1 

\\\ 

1 

% 

1 

1    1 
1    1 

1 

1 
t 

n 

, 

9 1 

1 
1 

15; 

f 

,'21 123 

/     !l 

V' 

3  15  17  jl9 

21   23 

-1 

0      - 

9      - 

8       - 

7      - 

6      - 

5      - 

4      - 

3      - 

2      - 

1 

0 

1      : 

3 

3 

- 

i        5 

6 

7 

3         9 

y-by 

Fig.  8(e)  —  y  versus  <p  —  by  for  QC  =  0.4,  k  =  2.5,  b  for  in  =  0.67ui(max)  and 
C  =  0.1  (Case  21). 

are  decelerated  by  the  circuit  field.  At  the  input  end  of  the  tube,  elec- 
trons are  uniformly  distributed  both  in  the  accelerating  and  decelerating 
field  regions.  Bunching  takes  place  when  the  accelerated  electrons  push 
forward  and  the  decelerated  ones  press  backward.  The  center  of  a  bunch 
of  electrons  is  located  well  inside  the  decelerating  field  region  because 
the  circuit  wave  travels  slower  than  the  electrons  on  the  average  (6  is 
positive).  The  effectiveness  of  the  amplification,  or  more  specifically  the  ! 
saturation  efficiency,  therefore  depends  on  (1),  how  tight  the  bunching :' 
is,  and  (2),  how  long  a  bunch  travels  inside  the  decelerating  field  region 
before  its  center  crosses  the  boundary  between  the  accelerating  and 
decelerating  fields. 

For  small-C,  the  ac  velocities  of  the  electrons  are  small  compared  with 
the  dc  velocity.  The  electron  bunch  stays  longer  with  the  decelerating 
circuit  field  before  reaching  the  saturation  level  when  h  or  QC  is  larger. 
On  the  other  hand,  the  space  charge  force,  or  large  QC  or  k  tends  to  dis- 
tort the  bunching.  As  the  consequence,  the  saturation  efficiency  increases  , 
with  h,  and  decreases  as  k  or  QC  increases.  When  C  becomes  finite  how- 


A   LARGE   SIGNAL  THEORY   OF  TRAVELING-WAVE   AMPLIFIERS       373 

ever,  the  ac  velocities  of  the  electrons  are  no  longer  small  as  compared 
I  with  their  average  speed.  The  velocity  spread  of  the  electrons  becomes 
,  an  important  factor  in  determining  the  efficiency.  Its  effect  is  to  loosen 
the  bunching,  and  consequently  it  lowers  the  saturation  level  and  re- 
duces the  limiting  efficiency.  It  is  seen  from  Figs.  5  and  6  that  the 
.   velocity  spread  increases  sharply  with  C  and  also  steadily  with  b  and  QC. 
\  This  explains  the  fact  that  in  the  present  calculation  the  saturation 
Eff./C  decreases  with  C  and  is  almost  constant  with  h  whereas  in  the 
1 1  small-C  theory  it  is  constant  with  C  and  increases  steadily  with  b. 

12.    ACKNOWLEDGEMENTS 

The  writer  wishes  to  thank  J.  R.  Pierce  for  his  guidance  during  the 
course  of  this  research,  and  L.  R.  Walker  for  many  interesting  discus- 
sions concerning  the  working  equations  and  the  method  of  calculating 

I  the  backward  wave.  The  writer  is  particularly  grateful  to  Miss  D.  C. 
Leagus  who,  under  the  guidance  of  V.  M.  Wolontis,  has  carried  out  the 

^  numerical  work  presented  with  endless  effort  and  enthusiasm. 

APPENDIX 

The  initial  conditions  at  i/  =  0  are  computed  from  Pierce's  linearized 
theory.  For  small-signal,  we  have 

ai(?/)  =  4A(y)  cos  (6  -f  ^2)2/  (A-1) 

«2(2/)  =  -4A(y)  sin  (6  +  ju2)y  (A-2) 

A(y)  =  ee"''  (A-3) 

Here  e  is  taken  equal  to  0.03,  a  value  which  has  been  used  in  Tien-Walker- 
'  Wolontis'  paper.  Define 

;  ^  =  wiy,  <po)  (A-4)  'X  =  pe-^'"  +  p*e^'^'>        (A-5) 

dy 

where  p*  is  the  conjugate  of  p.  After  substituting  (A-1)  to  (A-5)  into  the 
working  equations  (15)  to  (18)  and  carrying  out  considerable  algebraic 
work,  we  obtain  exactly  Pierce's  equation. 

2  (1  +  jC/i)(l  +  bC)  innn    \    ah  \^       r\  r\ 

(j  -  >iCfi  -h  j}/ibC)(ti  +  jb) 


provided  that 


+  CO 

—k\((>(.y  ,<po+<t>)—<p(.y  ,Vo)l['^+Cw(.y  ,ipo+(t>)] 
0 


(A-7) 
•  di^  sgn  (^(?/,  .i?o  +  «/))  -  9?(^,  <Po))  =  8eQC 

(1  -f  3Cy){ii  ^  jb)  I  e''"  cos  (arg  [(1  -f  jCm)(m  +  jb)]  +  my  -  ^0) 


374 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


Here  ^  =  Mi  +  JM2  or  Pierce's  rri  +  jiji  .  From  (A-7)  the  value  of  Up  is 
determined  for  a  given  QC.  The  ac  velocities  of  the  electrons  are  derived 
from  (A-4),  such  as, 


=    -26 


M 


M  +  jb 
1  +  jcn 


e"^"  cos  (  arg 


M 


M  +  jb 


rvi+iCM/j 


+  M22/  —  <Po 


(A-8) 


(A-1),  (A-2),  (A-7)  and  (A-8)  are  the  expressions  used  to  calculate  the 
initial  conditions  at  y  =  0,  Avhen  fn  and  jU2  are  solved  from  Pierce's  equa- 
tion (A-6). 

From  (12c),  the  particular  solution  of  the  backward  wave  at  small- 
signal  is  found  to  be 

j^,     .  .,     -2iC(l+iC/x)(M+ib) 


^Ml!/ 


2j  —  CfjL  -\-  icb 

r        [-2jC{\  - 


cos 


+  iCM)(M+i6)' 


Cn  +  jcb 


+  M2y  —  ^0 


which  agrees  with  Pierce's  analysis 


17 


3. 

4. 


REFERENCES 

1.  J.  R.  Pierce,  Traveling-Wave  Tubes,  D.  Van  Nostrand  Co.,  N.Y.,  1950,  p.  160. 

2.  R.  L.  Hess,  Some  Results  in  the  Large-Signal  Analysis  of  Traveling-Wave 

Tubes,  Technical  Report  Series  No.  60,  Issue  No.  131,  Electronic  Research 
Laboratory,  University  of  California,  Berkeley,  California. 

C.  K.  Birdsall,  unpublished  work. 

J.  J.  Caldwell,  unpublished  work. 

5.  P.  Parzen,  Nonlinear  Effects  in  Traveling-Wave  Amplifiers,  TR/AF-4,  Radia- 

tion Laboratory,  The  Johns  Hopkins  University,  April  27,  1954. 

6.  A.  Kiel  and  P.  Parzen,  Non-linear  Wave  Propagation  in  Traveling-Wave 

Amplifiers,  TR/AF-13,  Radiation  Laboratory,  The  Johns  Hopkins  Univer- 
sity, March,  1955. 

7.  A.  Nordsieck,  Theory  of  the  Large-Signal  Behavior  of  Traveling-Wave  Ampli- 
fiers, Proc.  I.R.E.,  41,  pp.  630-637,  May,  1953. 

H.  C.  Poulter,  Large  Signal  Theory  of  the  Traveling-Wave  Tube,  Tech.  Re- 
port No.  73,  Electronics  Research  Laboratory,  Stanford  University,  Cali- 
fornia, Jan.,  1954. 

P.  K.  Tien,  L.  R.  Walker  and  V.  M.  Wolontis,  A  Large  Signal  Theory  of  Trav- 
eling-Wave Amplifiers,  Proc.  LR.E.,  43,  pp.  260-277  March,  1955. 

J.  E.  Rowe,  A  Large  Signal  Analysis  of  the  Traveling-Wave  Amplifier,  Tech. 
Report  No.  19,  Electron  Tube  Laboratory,  University  of  Michigan,  Ann 
Arbor,  April,  1955. 
11.  P.  K.  Tien  and  L.  R.  Walker,  Correspondence  Section,  Proc.  I.R.E.,  43, 
p.  1007,  Aug.,  1955. 

Nordsieck,  op.  cit.,  equation  (1). 

L.  Brillouin,  The  Traveling-Wave  Tube  (Discussion  of  Waves  for  Large 
Amplitudes),  J.  Appl.  Phys.,  20,  p.  1197,  Dec,  1949. 

Pierce,  op.  cit.,  p.  9. 

Nordsieck,  op.  cit.,  equation  (4). 

Pierce,  op.  cit.,  equation  (7.13). 
17.  J.  R.  Pierce,  Theory  of  Traveling-Wave  Tube,  Appendix  A,  Proc.  I.R.E. 
35,  p.  121,  Feb.,  1947. 


8. 


10 


12 
13 

14 
15 
16 


A  Detailed  Analysis  of  Beam  Formation 
with  Electron  Guns  of  the  Pierce  Type 

By  W.  E.  DANIELSON,  J.  L.  ROSENFELD,*  and  J.  A.  SALOOM 

(Manuscript  received  November  10,  1955) 

The  theory  of  Cutler  and  Hines  is  extended  in  this  paper  to  permit  an 
analysis  of  heam-spreading  in  electron  guns  of  high  convergence.  A  lens 
correction  for  the  finite  size  of  the  anode  aperture  is  also  included.  The  Cutler 
and  Hines  theory  was  not  applicable  to  cases  where  the  effects  of  thermal 
velocities  are  large  compared  with  those  of  space  charge  and  it  did  not  include 
a  lens  correction.  Gun  design  charts  are  presented  which  include  all  of  these 
effects.  These  charts  may  he  conveniently  used  in  choosing  design  parameters 
to  produce  a  prescribed  beam. 

CONTENTS 

1 .  Introduction 377 

2.  Present  Status  of  Gun  Design;  Limitations 378 

3.  Treatment  of  the  Anode  Lens  Problem 379 

A.  Superposition  Approach 379 

B.  Use  of  a  False  Cathode 382 

C.  Calculation  of  Anode  Lens  Strength  by  the  Two  Methods 383 

4.  Treatment  of  Beam  Spreading,  Including  the  Effect  of  Thermal  Electrons  388 

A.  The  Gun  Region 388 

B.  The  Drift  Region 392 

5.  Numerical  Data  for  Electron  Gun  and  Beam  Design 402 

A.  Choice  of  Variables 402 

B.  Tabular  Data 402 

C.  Graphical  Data,  Including  Design  Charts  and  Beam  Profiles 402 

D.  Examples  of  Gun  Design  Using  Design  Charts 403 

6.  Comparison  of  Theory  with  Experiment 413 

A.  Measurement  of  Current  Densities  in  the  Beam 413 

B.  Comparison  of  the  Experimentally  Measured  Spreading  of  a  Beam  with 
that  Predicted  Theoretically 416 

C.  Comparison  of  Experimental  and  Theoretical  Current  Density  Distri- 
butions where  the  Minimum  Beam  Diameter  is  Reached 418 

D.  Variation  of  Beam  Profile  with  T 418 

7.  Some  Additional  Remarks  on  Gun  Design 418 


*  Mr.  Rosenfeld  participated  in  this  work  while  on  assignment  to  the  Labora- 
tories as  part  of  the  M.I.T.  Cooperative  Program. 

375 


376  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 

GLOSSARY   OF   SYMBOLS 

Ai ,  2  anode  designations 

B,  C  anode  potentials 

Ci ,  2  functions  used  in  evaluating  cr+' 

dA  increment  of  area 

dl,  dz  increments  of  length 

e  .  electronic  charge,  base  of  natural  logarithms 

En  electric  field  normal  to  electron  path 

F  modified  focal  length  of  the  anode  lens 

Fd  focal  length  of  the  anode  lens  as  given  by  Davisson^ 

Fn  force  acting  normal  to  an  electronic  path 

Fr ,  a  fraction  of  the  total  current  which  would  flow  through 

a  circle  of  radius  r,  a 

/,  Id  total  beam  current 

It  beam  current  within  a  radius,  r,  of  the  center 

J  current  density 

k  Boltzman's  constant 

K    -  a  quantity  proportional  to  gun  perveance 

m  electronic  mass 

P  gun  perveance 

P{r)  probability  that  a  thermal  electron  has  a  radial  posi- 

tion between  r  and  r  -\-  dr 

r  radial  distance  from  beam  axis 

Va  ,  c  anode,  cathode  radii 

r^  distance  from  beam  axis  to  path  of  an  electron  emitted 

with  zero  velocity  at  the  edge  of  the  cathode 

rgs  radius  of  circle  through  which  95%  of  the  beam  cur- 

rent would  pass 

f  distance  from  center  of  curvature  of  cathode;  hence, 

fc  is  the  cathode  radius  of  curvature  and  (fc  —  fa) 
is  the  distance  from  cathode  to  anode 

re+'  slope  of  edge  nonthermal  electron  path  on  drift  side  of 

enode  lens 

Te-'  slope  of  edge  nonthermal  electron  path  on  gun  side  of 

anode  lens 

R  a  dummy  integration  variable 

t  time 

T  cathode  temperature  in  degrees  K 

u  longitudinal  electron  velocity 

Vc ,  X  ,  y  transverse  electron  velocities 

V,  Va  ,  f  ,  X         beam  voltages  with  cathode  taken  as  ground 


BEAM    FORMATION    WITH    ELECTRON    GUNS  377 

V(f,    /■),    Vc.(f,     potential  distributions  used  in  the  anode  lens  study 
r),  etc. 

V'  voltage  gradient 

z  distance  along  the  beam  from  the  anode  lens 

2n,in  distance  to  the  point  where  rgs  is  a  minimum 

(  —  a)  Langmuir  potential  parameter  for  spherical  cathode- 

anode  gun  geometry 

7  slope  of  an  electron's  path  after  coming  into  a  space 

charge  free  region  just  beyond  the  anode  lens 

r  the  factor  which  divides  Fd  to  give  the  modified  anode 

focal  length 

5  dimensionless  radius  parameter 
€o                            dielectric  constant  of  free  space 
f  dimensionless  voltage  parameter 

6  slope  of  an  electron's  path  in  the  gun  region 
r}  charge  to  mass  ratio  for  the  electron 

fx  normalized  radial  position  in  a  beam 

a  the  radial  position  of  an  electron  which  left  the  cathode 

center  with  "normal"  transverse  velocity 
(T+'  slope  of  o--electron  on  drift  side  of  anode  lens 

a  J  slope  of  (T-electron  on  gun  side  of  anode  lens 

^  electric  flux 

1.   INTRODUCTION 

During  the  past  few  years  there  have  been  several  additions  to  the 
family  of  microwave  tubes  rec}uiring  long  electron  beams  of  small  diame- 
ter and  high  current  density.  Due  to  the  limited  electron  current  which 
can  be  "drawn  from  unit  area  of  a  cathode  surface  with  some  assurance 
of  long  cathode  operating  life,  high  density  electron  beams  have  been 
produced  largely  through  the  use  of  convergent  electron  guns  which 
increase  markedly  the  current  density  in  the  beam  over  that  at  the 
cathode  surface. 

An  elegant  approach  to  the  design  of  convergent  electron  guns  was 
provided  by  J.  R.  Pierce^  in  1940.  Electron  guns  designed  by  this  method 
are  known  as  Pierce  guns  and  have  found  extensive  use  in  the  produc- 
tion of  long,  high  density  beams  for  microwave  tubes. 

]\Iore  recent  studies,  reviewed  in  Section  2,  have  led  to  a  better  under- 
standing of  the  influence  on  the  electron  beam  of  (a)  the  finite  velocities 
with  which  electrons  are  emitted  from  the  cathode  surface,  and  (b)  the 
defocusing  electric  fields  associated  with  the  transition  from  the  ac- 
celerating region  of  the  gun  to  the  drift  region  beyond.  Although  these 
two  effects  have  heretofore  been  treated  separately,  it  is  in  many  cases 


378  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 

necessary  to  produce  electron  beams  under  circumstances  where  both 
effects  are  important  and  so  must  be  dealt  with  simultaneously  and  more 
precisely  than  has  until  now  been  possible.  It  is  the  purpose  of  this  paper 
to  provide  a  simple  design  procedure  for  typical  Pierce  guns  which  in- 
cludes both  effects.  Satisfactory  agreement  has  been  obtained  between 
measured  l^eam  contours  and  those  predicted  for  several  guns  having 
per\'eances  (i.e.,  ratios  of  beam  current  to  the  ^^  power  of  the  anode 
voltage)  from  0.07  X  10-«  to  0.7  X  10"^  amp  (volt)-3/2. 

2.    PRESENT   STATUS    OF   GUN    DESIGN  —  LIMITATIONS 

Gun  design  techniques  of  the  type  originally  suggested  by  J.  R.  Pierce 
were  enlarged  in  papers  by  SamueP  and  by  Field^  in  1945  and  1946. 
Samuel's  work  did  not  consider  the  effect  of  thermal  velocities  on  beam 
shape  and,  although  Field  pointed  out  the  importance  of  thermal  veloci- 
ties in  limiting  the  theoretically  attainable  current  density,  no  method 
for  predicting  beam  size  and  shape  by  including  thermal  effects  was 
suggested.  The  problem  of  the  divergent  effect  of  the  anode  lens  was 
treated  in  terms  of  the  Davisson"*  electrostatic  lens  formula,  and  no 
corrections  were  applied.* 

More  recently.  Cutler  and  Hines^  and  also  Cutler  and  Saloom^  have 
presented  theoretical  and  experimental  work  which  shows  the  pro- 
nounced effects  of  the  thermal  velocity  distribution  on  the  size  and  shape 
of  beams  produced  by  Pierce  guns.  Cutler  and  Saloom  also  point  to  the 
critical  role  of  the  beam-forming  electrode  in  minimizing  beam  distor- 
tion due  to  improper  fields  in  the  region  where  the  cathode  and  the 
beam-forming  electrode  would  ideally  meet.  With  regard  to  the  anode 
lens  effect,  these  authors  also  show  experimental  data  which  strongly 
suggest  a  more  divergent  lens  than  given  by  the  Davisson  formula.  The 
Hines  and  Cutler  thermal  velocity  calculations  have  been  used"'  "^  to 
predict  departures  in  current  density  from  that  which  should  prevail  in 
ideal  beams  where  thermal  electrons  are  absent.  Their  theory  is  limited, 
however,  by  the  assumption  that  the  beam-spreading  caused  by  thermal 
velocities  is  small  compared  to  the  nominal  beam  size. 

In  reviewing  the  various  successes  of  the  above  mentioned  papers  in 
affording  valuable  tools  for  electron  beam  design,  it  appeared  to  the 
present  authors  that  significant  improvement  could  be  made,  in  two 
respects,  by  extensions  of  existing  theories.  First,  a  more  thorough  in- 


*  It  is  in  fact  erroneously  statoci  in  Reference  5  that  the  lens  action  of  an  actual 
structure  must  be  somewhat  weaker  than  i)re(licted  by  the  Davisson  formula  so 
that  the  beam  on  leaving  the  anode  hole  is  more  convergent  than  would  be  calcu- 
lated by  llie  Davisson  method.  This  cjuestion  is  discussed  further  in  Section  3. 


BEAM    FORMATION    WITH    ELECTRON    GUNS  379 

vestigation  of  the  anode  lens  effect  was  called  for;  and  second,  there  was 
a  need  to  extend  thermal  velocity  calculations  to  include  cases  where 
the  percentage  increase  in  beam  size  due  to  thermal  electrons  was  as 
large  as  100  per  cent  or  200  per  cent.  Some  suggestions  toward  meeting 
this  second  need  have  been  included  in  a  paper  by  M.  E.  Hines.*  They 
have  been  applied  to  two-dimensional  beams  by  R.  L.  Schrag.^  The 
particular  assumptions  and  methods  of  the  present  paper  as  applied  to 
the  two  needs  cited  above  are  somewhat  different  from  those  of  Refer- 
ences 8  and  9,  and  are  fully  treated  in  the  sections  which  follow. 

3.  TREATMENT  OF  THE  ANODE  LENS  PROBLEM 

Using  thermal  velocity  calculations  of  the  type  made  in  Reference  6, 
it  can  easily  be  shown  that  at  the  anode  plane  of  a  typical  moderate 
perveance  Pierce  type  electron  gun,  the  average  spread  in  radial  posi- 
tion of  those  electrons  which  originate  from  the  same  point  of  the  cathode 
is  several  times  smaller  than  the  beam  diameter.  For  guns  of  this  type, 
then,  we  may  look  for  the  effect  of  the  anode  aperture  on  an  electron 
beam  for  the  idealized  case  in  which  thermal  velocities  are  absent  and 
confidently  apply  the  correction  to  the  anode  lens  formula  so  obtained 
to  the  case  of  a  real  beam. 

Several  authors  have  been  concerned  with  the  diverging  effect  of  a 
hole  in  an  accelerating  electrode  where  the  field  drops  to  zero  in  the 
space  beyond, ^°  but  these  treatments  do  not  include  space  charge  effects 
except  as  given  by  the  Davisson  formula  for  the  focal  length,  Fd  ,  of 
the  lens: 

F.  =  -^  (1) 

where  V  would  be  the  magnitude  of  the  electric  field  at  the  aperture  if 
it  were  gridded,  and  V  would  be  the  voltage  there. 

In  attempting  to  describe  the  effect  of  the  anode  hole  with  more  ac- 
curacy than  (1)  affords,  we  have  combined  analytical  methods  with 
electrolytic  tank  measurements  in  two  i-ather  different  ways.  The  first 
method  to  be  given  is  more  rigorous  than  the  second,  hut  a  modification 
of  the  second  method  is  much  easier  to  use  and  gives  essentially  the 
same  result. 

A.  Siipcrposition  Approach  to  the  Anode  Lens  Problem 

Special  techniques  are  required  for  finding  electron  trajectories  in  a 
space  charge  limited  Pierce  gun  having  a  non-gridded  anode.  M.  E. 


380  THE    BELL    SYSTEM    TECHNICAL   JOURNAL,    MARCH    1956 

Hines  has  suggested*  that  a  fairly  accurate  description  of  the  potential 
distribution  in  such  guns  can  be  obtained  by  a  superposition  method  as 
follows: 

By  the  usual  tank  methods,  find  suitable  beam  forming  electrode  and 
anode  shapes  for  conical  space  charge  limited  flow  in  a  diode  having! 
cathode  and  anode  radii  of  curvature  given  by  fc  and  f„i  ,  respectively, 
as  shown  in  Fig.  1(a).  Using  the  electrolytic  tank  with  an  insulator  along 
the  line  which  represents  the  beam  edge,  trace  out  an  equipotential 
which  intersects  the  insulator  at  a  distance  fa2  from  the  cathode  center 
of  curvature.  Let  the  cathode  be  at  ground  potential  and  let  the  voltage 
on  anode  Ai  be  called  B.  Suppose,  now,  that  we  are  interested  in  electron 
trajectories  in  a  non-gridded  gun  where  the  edge  of  the  anode  hole  is  a 
distance  fai  from  the  center  of  curvature  of  the  cathode.  Let  the  voltage, 
C,  for  this  anode  be  chosen  the  same  as  the  value  of  the  equipotential 
traced  out  above  for  the  case  of  cathode  at  ground  potential  and  A\ 
at  potential  B.  If  we  consider  the  space  charge  limited  flow  from  a 
cathode  which  is  followed  by  the  apertured  anode,  Ai  ,  and  the  full 
anode,  Ai ,  at  potentials  C  and  B,  respectively,  it  is  clear  that  a  conical 
flow  of  the  type  which  would  exist  between  concentric  spheres  will  re- 
sult. The  flow  for  such  cases  was  treated  by  Langmuir,^  and  the  associ- 
ated potentials  are  commonly  called  the  "Langmuir  potentials." 

If  we  operate  both  Ai  and  A2  at  potential  C,  however,  the  electrons 
will  pass  through  the  aperture  in  anode  A2  into  a  nearly  field-free  region. . 
If  the  distance,  fa2  —  Tai  ,  from  A2  to  Ai  is  greater  than  the  diameter  of 
the  aperture  in  A2 ,  the  flow  will  depend  very  little  on  the  shape  of  Ai 
and  the  electron  trajectories  and  associated  equipotentials  will  be  of  the 
type  we  wish  to  consider  except  in  a  small  region  near  Ai  .  We  will  shortly 
make  use  of  the  fact  that  the  space  charge  between  cathode  and  A 2  is 
not  changed  much  when  the  voltage  on  Ai  is  changed  from  B  to  C,  but 
first  we  will  define  a  set  of  potential  functions  which  will  be  needed. 

In  order  to  obtain  the  potential  at  arbitrary  points  in  any  axially  sym- 
metric gun  when  space  charge  is  not  neglected,  w^e  may  superpose  po- 
tential solutions  to  3  separate  problems  where,  in  each  case,  the  boundary 
condition  that  each  electrode  be  an  equipotential  is  satisfied.  We  will 
follow  the  usual  notation  in  using  f  for  the  distance  of  a  general  point 
from  the  cathode  center  of  curvature,  and  r  for  its  radial  distance  from 
the  axis  of  symmetry.  Let  Vdr,  r),  Vh(r,  >')  and  Vsdr,  r)  be  the  three 
potential  solutions  where:  (1)  Vaif,  r)  is  the  solution  for  the  case  of  no 
space  charge  with  Ai  and  cathode  at  zero  potential  and  A  2  at  potential 
C,  (2)  Vb{r^  r)  is  the  solution  for  the  case  of  no  space  charge  with  A2 


*  Verbal  disclosure. 


BEAM    FORMATION    WITH    ELECTRON    GUNS 


381 


and  cathode  at  zero  potential  and  Ai  at  potential  B,  and  (3)  Vsc(f,  r)  is 
the  soUition  when  space  charge  is  present  but  when  Ax  ,  A^  ,  and  cathode 
are  all  grounded. 

If  the  configuration  of  charge  which  contributes  to  Vs<-(f,  r)  is  that 
corresponding  to  ideal  Pierce  type  flow,  then  we  can  use  the  principle 
of  superposition  to  give  the  Langmuir  potential,  VL(r,  r): 


VUr,  r)  =  Vcif,  r)  +  V,{f,  r)  +  V..{f,  r) 


(2) 


Furthermore,  the  potential  configuration  for  the  case  where  ^i  and  A2 
are  at  potentical  C  can  be  written 


V  =V.-\-^V,  +  F(.c)' 


(3) 


where  the  functional  notation  has  been  dropped  and  F(sc)'  is  the  po 
icntial  due  to  the  new  space  charge  when  Ai  and  A2  are  grounded. 
We  are  now  ready  to  use  the  fact  that  F(sc)'  may  be  well  approximated 
1)3'  Fsc  which  is  easily  obtained  from  (2).  This  substitution  may  be 
justified  by  noting  that  the  space  charge  distribution  in  a  gun  using  a 
\'oltage  C  for  Ai  does  not  differ  significanth^  from  the  corresponding  dis- 
tribution when  Ai  is  at  voltage  B  except  in  the  region  near  and  beyond 
A-i  where  the  charge  density  is  small  anyway  (because  of  the  high  electron 
velocities  there).  Substituting  Fsc  as  given  by  (2)  for  F(sc)'  in  (3)  then 
gives 


V 


Vi 


1 


B, 


V, 


(4) 


We  have  thus  obtained  an  expression,  (4),  for  the  potential  at  an  arbi- 


ANODE  A2 

v=c 


ANODE  A, 
V  =  B 


CATHODE 


Fig.  1(a)  — ■  Electrode  configuration  for  anode  lens  evaluation  in  Section  2>A. 


382  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    1956 

■i 

trary  point  in  our  gun  in  terms  of  the  well  known  solution  for  space 
charge  limited  flow  between  two  concentric  spheres,  Vl  ,  and  a  potential 
distribution,  Vb ,  which  does  not  depend  on  space  charge  and  can  there- 
fore be  obtained  in  the  electrolytic  tank.  Once  the  potential  distribution 
is  found,  electron  trajectories  may  be  calculated,  and  an  equivalent  lens 
sj^stem  found.  Equation  (4)  is  used  in  this  way  in  Part  C  as  one  basis  for 
estimating  a  correction  to  the  Davisson  equation.  (It  will  be  noted  that  i 
(4)  predicts  a  small  but  finite  negative  field  at  the  cathode.  This  is  be- 
cause the  space  charge  density  associated  with  Fsc  is  slightly  greater 
near  the  cathode  than  that  associated  with  F(sc)'  ,  and  it  is  this  latter 
space  charge  which  will  make  the  field  zero  at  the  cathode  under  real 
space  charge  limited  operation.  Equation  (4),  as  applied  in  Part  C  of  this 
section,  is  used  to  give  the  voltage  as  a  function  of  position  at  all  points 
except  near  the  cathode  where  the  voltage  curves  are  extended  smoothly 
to  make  the  field  at  the  cathode  vanish.) 

B.   Use  of  a  False  Cathode  in  Treating  the  Anode  Lens  Problem 

Before  evaluating  the  lens  effect  by  use  of  (4),  it  will  be  useful  to  de- 
velop another  approach  which  is  a  little  simpler.  The  evaluation  of  the 
lens  effect  predicted  by  both  methods  will  then  be  pursued  in  Part  C 
where  the  separate  results  are  compared. 

In  Part  A  we  noted  that  no  serious  error  is  made  in  neglecting  the  dif- 
ference between  the  two  space  charge  configurations  considered  there 
because  these  differences  were  mainly  in  the  very  low  space  charge 
region  near  and  beyond  A2  .  It  similarly  follows  that  we  can,  with  only  1 
a  small  decrease  in  accuracy,  ignore  the  space  charge  in  the  region  near 
and  beyond  A2  so  long  as  we  properly  account  for  the  effect  of  the  high 
space  charge  regions  closer  to  the  cathode.  To  place  the  foregoing  obser- 
vations on  a  more  quantitative  basis,  we  may  graph  the  Langmuir  po- 
tential (for  space  charge  limited  flow  between  concentric  spheres)  versus 
the  distance  from  cathode  toward  anode,  and  then  superpose  a  plot  of 
the  potential  from  LaPlace's  equation  (concentric  spheres;  no  space 
charge)  which  will  have  the  same  value  and  slope  at  the  anode.  The  La- 
Place  curve  will  depart  significantly  from  the  Langmuir  in  the  region  of 
the  cathode,  but  will  adequately  represent  it  farther  out."  Our  experi- 
ence has  shown  that  the  representation  is  "adequate"  until  the  difference 
between  the  two  potentials  exceeds  about  2  per  cent  of  the  anode  voltage. 
Then,  since  space  charge  is  not  important  in  the  region  near  the  anode 
for  the  case  of  a  gridded  Pierce  gun,  corresponding  to  space  charge 
limited  flow  between  concentric  spheres,  it  can  be  expected  to  be  similarly 
unimportant  for  cases  where  the  grid  is  replaced  by  an  aperture.  Let  us 


I 


BEAM   FORMATION   WITH    ELECTRON    GUNS 


383 


therefore  consider  a  case  where  electrons  are  emitted  perpendicularly 
and  with  finite  velocity  from  what  would  be  an  appropriate  spherical 
equipotential  between  cathode  and  anode  in  a  Pierce  type  gun.  So  long 
as  (a)  there  is  good  agreement  between  the  LaPlace  and  Langmuir  curves 
at  this  artificial  cathode  and  (b)  the  distance  from  this  artificial  cathode 
to  the  anode  hole  is  somewhat  greater  than  the  hole  diameter,  we  will 
liiid  that  the  divergent  effect  of  the  anode  hole  will  be  very  nearly  the 
same  in  this  concocted  space  charge  free  case  as  in  the  actual  case  where 
space  charge  is  present.  (The  quantitative  support  for  this  last  state- 
ment comes  largely  from  the  agreement  between  calculations  based  on 
this  method  and  calculations  by  method  A.)  The  electrode  configura- 
tion is  shown  in  Fig.  1(b),  and  the  potential  distribution  in  this  space 
charge  free  anode  region  can  now  be  easily  obtained  in  the  electrolytic 
j  tank.  This  potential  distribution  will  be  used  in  the  next  section  to  pro- 
^•ide  a  second  basis  for  estimating  a  correction  to  the  Davisson  equation. 

C.  Calculation  of  Anode  Lens  Strength  by  the  Two  Methods 

The  Davisson  equation,  (1),  may  be  derived  by  assuming  that  none 
of  the  electric  field  lines  which  originate  on  charges  in  the  cathode-anode 
region  leave  the  beam  before  reaching  the  ideal  anode  plane  where  the 
voltage  is  F,  and  that  all  of  these  field  lines  leave  the  beam  symmetrically 
and  radially  in  the  immediate  neighborhood  of  the  anode.  Electrons 
I  are  thus  considered  to  travel  in  a  straight  line  from  cathode  to  anode, 
and  then  to  receive  a  sudden  radial  impulse  as  they  cross  radially  diverg- 
ing electric  field  lines  at  the  anode  plane.  A  discontinuous  change  in 


CATHODE 


ANODE  A2 
V  =  C 


ANODE  A, 

v  =  c 


(b) 


^  FALSE 
CATHODE 


Fig.  1(b)  —  The  introduction  of  a  false  cathode  at  the  appropriate  potential 
lUows  the  effect  of  space  charge  on  the  potential  near  the  anode  hole  to  be  satis- 
:ictorily  approximated  as  discussed  in  Section  3i?. 


384  THE    BELL    SYSTEM    TECHNICAL   JOURNAL,    MARCH    1956 

slope  is  therefore  produced  as  is  common  to  all  thin  lens  approximations. 
The  diverging  effect  of  electric  field  lines  which  originate  on  charges 
which  have  passed  the  anode  plane  is  then  normally  accounted  for  by 
the  universal  beam  spread  curve/"  In  our  attempt  to  evaluate  the  lens 
effect  more  accurately,  we  will  still  depend  upon  using  the  universal 
beam  spread  curve  in  the  region  following  the  lens  and  on  treating  the ; 
equivalent  anode  lens  as  thin.  Consequently  our  improved  accuracy 
must  come  from  a  mathematical  treatment  which  allows  the  electric 
field  lines  originating  in  the  cathode-anode  region  to  leave  the  beam  grad- 
ually, rather  than  a  treatment  where  all  of  these  flux  lines  leave  the  beam  , 
at  the  anode  plane.  In  practice  the  measured  perveances,  P(=  I/V^'^), 
of  active  guns  of  the  type  considered  here  have  averaged  within  1  or  2 
per  cent  of  those  predicted  for  corresponding  gridded  Pierce  guns.  There- 
fore the  total  space  charge  between  cathode  and  anode  is  much  the 
same  with  and  without  the  use  of  a  grid,  even  though  the  charge  dis- 
tribution is  not  the  same  in  the  two  cases.  The  total  flux  which  must 
leave  our  beam  is  therefore  the  same  as  that  which  will  leave  the  cor- , 
responding  idealized  beam  and  we  may  write 

yp    =      I    EndA     =    TT/VFidea/  (5) 

w^here  En  is  the  electric  field  normal  to  the  edge  of  the  beam,  ra  =  rdfa/fc) 
is  the  beam  radius  at  the  anode  lens,  and  Videai  is  the  magnitude  of  the 
field  at  the  corresponding  gridded  Pierce  gun  anode. 

To  find  the  appropriate  thin  lens  focal  length  we  will  now  find  the 
total  integrated  transverse  impulse  which  would  be  given  to  an  elec- 
tron which  follows  a  straight-line  path  on  both  sides  of  the  lens  (see  Fig. 
2),  and  we  will  equate  this  impulse  to  wAw  where  An  is  the  transverse 
velocity  given  to  the  electron  as  it  passes  through  the  equivalent  thin 
lens.  In  this  connection  we  will  restrict  our  attention  to  paraxial  elec- 
trons and  evaluate  the  transverse  electric  fields  from  (4)  and  from  the 
tank  plot  outlined  in  Section  B,  respectively.  The  total  transverse  im- 
pulse experienced  by  an  electron  can  be  written 

f      Fn  dt  =  e  [      —dl  (()) 

J  Path  J  Path     U 

where  u  is  the  velocity  along  the  path  and  Fn  is  the  force  normal  to  the 
path. 

We  will  usually  find  that  the  correction  to  (1)  is  less  than  about  20 
per  cent.  It  will  therefore  be  worthwhile  to  put  (6)  in  a  form  which  in 
effect  allows  us  to  calculate  deviaiions  from  Fu  as  given  by  (1)  instead 


BEAM    FORMATION    WITH    ELECTRON    GUNS 


385 


1  of  deriving  a  completely  new  expression  for  F.  In  accomplishing  this  piir- 
f  pose,  it  will  be  helpful  to  define  a  dimensionless  function  of  radius,  6,  by 


-  =  1  +  5, 
r 

and  a  dimensionless  function  of  voltage,  f,  by 


(7a) 


(7b) 


where  Ta  is  the  radius  at  the  anode  lens  when  the  lens  is  considered  thin, 
and  T^'x  is  a  constant  voltage  to  be  specified  later.  (Note  that  the  quan- 
tities 5  and  f  are  not  necessarily  small  compared  to  1.)  Using  u  =  \/2r]V, 
and  substituting  for  -y/V  from  (7b)  we  obtain 


f  En  dl  4         r        , 

=  7~7tW  /  ^"^1  +  r  +  5  +  rs)  ^z 


(8) 


where  use  has  also  been  made  of  (7a)  in  the  form  1  =  r(l  +  d)/ra  .  Now, 
as  outlined  above,  we  equate  this  impulse  to  771  An,  and  we  obtain 


^»  =  WW.  (/  ''■'' ''  +  /  ''"'■'^  + '  +  ^''  'i 


(9) 


CATHODE 


Fig.  2  —  The  heavy  line  represents  an  electron's  path  when  the  effect  of  the 
.•mode  hole  may  be  represented  by  a  thin  lens,  and  when  space  charge  forces  are 
iihsent  in  the  region  following  the  anode  aperture.  For  paraxial  electrons,  the 
(negative)  focal  length  is  related  to  the  indicated  angles  by  (y  =  0  +  Ta/F). 


386 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


CENTER  OF 

~~  CURVATURE 

OF   CATHODE 

SURFACE 


Fig.  3  —  The  gun  parameters  used  in  Section  SC  for  comparing  two  methods  of 
evaluating  the  effect  of  the  anode  lens. 

The  first  integral  can  be  obtained  from  (5) ;  hence,  if  we  are  able  to  choose 
Vx  so  that  the  second  integral  vanishes,  we  may  write: 


Au  = 


raV'2riVx 


The  reciprocal  of  the  thin  lens  focal  length  is  therefore 

i  _       ^  _  ^' 

F  ~  ~raUf  ^  ~^VWf 


(10) 


where  w/  and  F/  are  the  final  velocity  and  voltage  of  the  electron  after 
it  leaves  the  lens  region. 

The  real  task,  then,  is  to  use  the  potential  distribution  in  the  gun  as 
obtained  by  the  methods  of  Part  A  or  Part  B  above  to  find  the  value  of 
V X  which  causes  the  last  integral  in  (9)  to  vanish :  To  compare  the  two 
focal  lengths  obtained  by  the  methods  of  Part  A  and  B  respectively,  a 
specific  tank  design  of  the  type  indicated  in  Fig.  1  was  carried  out.  The 
relevant  gun  parameters  are  indicated  in  Fig.  3.  Approximate  voltages 
on  and  near  the  beam  axis  were  obtained  as  indicated  in  Parts  A  and  B, 
above,  with  the  exception  that  in  the  superposition  method,  A,  special 
techniques  were  used  to  subtract  the  effect  of  the  space  charge  lying  in 
the  post-anode  region  (because  the  effect  of  this  space  charge  is  accounted 
for  separately  as  a  divergent  force  in  the  drift  region*).  From  these  data, 

*  See  Section  4B. 


BEAM  FOKMATION  WITH  ELECTRON  GUNS 


387 


800   805    810    815    820   825   830   835   840   845   850    855   860 


Fig.  4  —  Curves  for  finding  the  value  of  Fx  to  be  used  in  equation  (10)  for  the 
set  of  gun  parameters  of  Fig.  3. 


l)oth  the  direction  and  magnitude  of  the  total  electric  field  near  the 
beam  axis  were  (with  much  labor)  determined.  Once  these  data  had 
been  obtained,  a  trial  value  was  selected  for  Vx  ,  and  the  corresponding 
local  length  was  calculated  by  (10).  This  enabled  the  electron's  path 
through  the  associated  thin  lens  to  be  specified  so  that,  at  this  point  in 
the  procedure,  both  r  and  V  were  known  functions  of  ^,  and  the  quan- 
tities 8  and  f  were  then  obtained  as  functions  of  €  from  (7).  Finally  the 
second  integral  in  (9)  was  evaluated  for  the  particular  Vx  chosen,  and 
then  the  process  was  repeated  for  other  values  of  Vx  .  Fig.  4  shows  curves 
whose  ordinates  are  proportional  to  this  second  integral  and  whose 
abscissae  are  trial  values  for  Vx  .  As  noted  above,  the  appropriate  value 
for  Vx  is  that  value  which  makes  the  ordinate  vanish,  so  that  we  obtain 
T'c  =  813  and  839  for  methods  A  and  B,  respectively.  The  percentage 
difference  in  the  focal  lengths  obtained  by  the  two  methods  is  thus  only 
1 .6  per  cent,  and  the  reasonableness  of  making  calculations  as  outlined 
in  Part  B  is  thus  put  on  a  more  quantitative  basis. 

Even  calculations  based  on  the  method  of  Part  B  are  tedious,  and  we 
naturally  look  for  simpler  methods  of  estimating  the  lens  effect.  In  this 
fonnection  we  have  found  that  Vx  is  usually  well  approximated  by  the 
\alue  of  the  potential  at  the  point  of  intersection  between  the  beam  axis 
and  the  ideal  anode  sphere.  The  specific  values  of  the  potential  at  this 
point  as  obtained  by  the  methods  of  Parts  A  and  B  were  814  and  827, 
respectively.  It  will  be  noted  that  these  values  agree  remarkably  well 
with  the  values  obtained  above.  Furthermore,  very  little  extra  effort  is 
required  to  obtain  the  potential  at  this  intersection  in  the  false  cathode 
case: 


I 

388  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    1956 

Electrolytic  tank  measurements  are  normally  made  in  the  cathode- 
anode  region  to  give  the  potential  variation  along  the  outside  edge  of 
the  electron  beam  (for  comparison  with  the  Langmuir  potential) ;  hence, 
by  tracing  out  a  suitable  equipotential  line,  the  shape  of  the  false  cathode 
can  easily  be  obtained.  With  the  false  cathode  in  place  and  at  the  proper 
potential,  the  approximate  value  for  Vx  is  then  obtained  by  a  direct  tank 
measurement  of  the  potential  at  an  axial  point  whose  distance  from  the 
true  cathode  center  is  (fc  —  fa)  as  outlined  above.  Although  finite  elec- 
tron emission  velocities  typically  do  not  much  influence  the  trajectory 
of  an  electron  at  the  anode,  they  do  nevertheless  significantly  alter  the 
beam  in  the  region  beyond.  It  is  in  this  affected  region  where  experi- 
mental data  can  be  conveniently  taken.  We  must,  therefore,  postpone  a 
comparison  of  lens  theory  with  experiment  until  the  effect  of  thermal 
velocities  has  been  treated.  At  that  time  theoretical  predictions  com- 
bining the  effects  of  both  thermal  velocities  and  the  anode  lens  can  be 
made  and  compared  with  experiment.  Such  a  comparison  is  made  in 
Section  6. 

4.    TREATMENT  OF  BEAM  SPREADING,  INCLUDING  THE  EFFECT  OF  THERMAL 
ELECTRONS 

Jn  Section  2  the  desirability  of  having  an  approach  to  the  thermal 
spreading  of  a  beam  which  would  be  applicable  under  a  wide  variety  of 
conditions  was  stressed.  In  particular,  there  was  a  need  to  extend  ther- 
mal velocity  calculations  to  include  the  effects  of  thermal  velocities  even 
when  electrons  with  high  average  transverse  velocities  perturb  the  beam 
size  by  as  much  as  100  or  200  per  cent.  Furthermore,  a  realistic  mathe- 
matical description  which  would  allow  electrons  to  cross  the  axis  seemed 
essential.  The  method  described  below  is  intended  adequately  to  answer 
these  requirements. 

A.  The  Gun  Region 

The  Hines-Cutler  method  of  including  the  effect  of  thermal  velocities 
on  beam  size  and  shape  leads  one  to  conclude  that,  for  usual  anode 
voltages  and  gun  perveance,  the  beam  density  profile  in  the  plane  of 
the  anode  hole  is  not  appreciably  altered  by  thermal  velocities  of  emis- 
sion. (This  statement  will  be  verified  and  put  on  a  more  quantitative 
basis  below.)  Under  these  conditions,  the  beam  at  the  anode  is  ade- 
quately described  by  the  Hines-Cutler  treatment.  We  will  therefore  find 
it  convenient  to  adopt  their  notation  where  possible,  and  it  will  be 
worthwhile  to  review  their  approach  to  the  thermal  problem. 


BEAM   FORMATION    WITH    ELECTRON   GUNS  389 

It  is  assumed  that  electrons  are  emitted  from  the  cathode  of  a  therm- 
ionic gun  with  a  IMaxwelhan  distribution  of  transverse  velocities 

ZTTfC  1 

where  Jc  is  the  cathode  current  density  in  the  z  direction,  T  is  the  cath- 
jode  temperature,  and  v^:  and  Vy  are  transverse  velocities.  The  number 
iof  electrons  emitted  per  second  with  radially  directed  voltages  between 

V  and  V  +  dV  is  then 


-(.Ve/kT) 


(S) 


^J.  =  /.e— -^^^(^^j  (12) 

Now  in  the  accelerating  region  of  an  ideal  Pierce  gun  (and  more  generally 
I  in  any  beam  exhibiting  laminar  flow  and  having  constant  current  density 
()\'er  its  cross  section)  the  electric  field  component  perpendicular  to  the 
axis  of  symmetry  must  vary  linearly  with  radius.  Conseciuently  Hines 
and  Cutler  measure  radial  position  in  the  electron  beam  as  a  fraction, 
^,  of  the  outer  beam  radius  (re)  at  the  same  longitudinal  position, 

r  =  fire  (13) 

The  laminar  flow  assumption  for  constant  current  densities  and  small 
beam  angles  implies  a  radius  of  curvature  for  laminar  electrons  which 
so  varies  linearly  with  radius  at  any  given  cross  section  so  that 


a 


Substituting  for  r  from  (13),  (14)  becomes 

rfV    ,    /2  dre\  dfj. 

d^^VcTt)dt=^  ^^^^ 

where  Ve  and  dr  /dt  can  be  easily  obtained  from  the  ideal  Langmuir 
solution.  Since  the  eciuation  is  linear  in  /x,  we  are  assured  that  the  radial 
position  of  a  non-ideal  electron  that  is  emitted  with  finite  transverse 
velocity  from  the  cathode  center  (where  ^  =  0)  will,  at  any  axial  point, 
be  proportional  to  dii/dt  at  the  cathode. 

Let  us  now  define  a  quantity  "o-"  such  that  n  =  a/re  is  the  solution 
to  (15)  with  the  boundary  conditions  /Xr  =  0  and 

_  1 
where  the  subscript  c  denotes  evaluation  at  the  cathode  surface,  k  is 


390  THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 

Boltzman's  constant,  T  is  the  cathode  temperature  in  degrees  Kelvin, 
and  m  is  mass  of  the  electron.  For  the  case  ixc  =  0,  but  with  arbitrary 
initial  transverse  velocity,  we  will  then  have 

/^\ 

^^nl_    /kf  ^^^'^ 

Tc  y     m 

Plence  we  can  express  a  in  terms  of  the  thermal  electron's  radial  po- 
sition (r),  and  its  initial  transverse  velocity,  Vc , 


y  m  _     y 


.  .     -  .  /kT 

dt    }  f 


The  quantity  a  can  now  be  related  to  the  radial  spread  of  thermal 
electrons  (emitted  from  a  given  point  on  the  cathode)  with  respect  to 
an  electron  with  no  initial  velocity:  By  (11)  we  see  that  the  number 
of  electrons  leaving  the  cathode  with  dji/dt  =  Vc/ve  is  proportional  to  Vc 
exp  —Vcm/2kT.  Suppose  many  experiments  were  conducted  where  all 
electrons  except  one  at  the  cathode  center  had  zero  emission  velocity, 
and  suppose  the  number  of  times  the  initial  transverse  velocity  of  the 
single  thermal  electron  were  chosen  as  Vc ,  is  proportional  to  Vc  exp 
—  Vcm/2kT.  Then  the  probability,  P{r),  that  the  thermal  electron 
would  have  a  radial  position  between  r  and  r  -\-  dr  when  it  arrived  at  the 
transverse  plane  of  interest  would  be  proportional  to  Vc  exp  —Vc^(m/2kT). 
Here  Vc  is  the  proper  transverse  velocity  to  cause  arrival  at  radius  r,  and 
by  (17)  we  have 

a   y     m 
so  that  the  probability  becomes 

Pir)  =  J.e-^^'''-'^  d  (^Q  (18) 

We  therefore  identify  cr  with  the  standard  deviation  in  a  normal  or 
Gaussian  distribution  of  points  in  two  dimensions.  At  the  real  cathod(\ 
thermal  electrons  are  simultaneously  being  emitted  from  the  cathode 
surface  with  a  range  of  transverse  velocities.  However,  if  a  as  definml 
above  is  small  in  comparison  with  r,. ,  the  forces  experienced  by  a  ther- 
mal electron  when  other  thermal  electrons  are  present  will  be  very  nearly 


BEAM    FORMATION    WITH    ELECTRON    GUNS 


391 


2.0 
1.8 
1.6 


>     1.4 
t     1.2 


\%y 


1.0 


0.8 


0.6 


0.4 


1.0 


1.2 


1.4 


1.6 


1.8       2.0       2.2       2.4 


2.6 


2.8        3.0        3.2       3.4       3.6       3.8        4.0 


Fig.  5  —  Curves  useful  in  finding  the  transverse  displacement  of  electron  tra- 
i  jectories  at  the  anode  of  Pierce-type  guns. 

i 

tlie  same  as  the  forces  involved  in  the  equations  above.  Thus  if  o-  <3C  J'e , 

(18)  may  be  taken  as  the  distribution,  in  a  transverse  plane,  of  those 
electrons  which  were  simultaneously  emitted  at  the  cathode  center. 
I  Furthermore,  the  nature  of  the  Pierce  gun  region  is  such  that  electrons 
emitted  from  any  other  point  on  the  cathode  will  be  similarly  distributed 
\\  ith  respect  to  the  path  of  an  electron  emitted  from  this  other  point 
w  ith  zero  transverse  velocity  (so  long  as  they  stay  within  the  confines 
,  of  the  ideal  beam).  Hines  and  Cutler  have  integrated  (15)  with  n^  =  0 
'  and  {dn/dt)c  =  1  to  give  g/  {fc\/kT/'2eV^  at  the  anode  as  a  function  of 
;  /",  /fo  .  This  relationship  is  included  here  in  graphical  form  as  Fig.  5. 
,      For  a  large  class  of  magnetically  shielded  Pierce-type  electron  guns, 
including  all  that  are  now  used  in  our  traveling  wave  tubes,  Ve/a  at  the 
anode  is  indeed  found  to  be  greater  than  5  (in  most  cases,  greater  than 
10)  so  that  evaluation  of  a  at  the  anode  of  such  guns  can  be  made  with 
considerable  accuracy  by  the  methods  outlined  above.  One  source  of 
error  lies  in  the  assumption  that  electrons  which  are  emitted  from  a 
point  at  the  cathode  edge  become  normally  distributed  about  the  cor- 
responding non-thermal  (no  transverse  velocity  of  emission)  electron's 
path,  and  with  the  same  standard  deviation  as  calculated  for  electrons 
from  the  cathode  center.  In  the  gun  region  where  Ve/a  tends  to  be  large 
this  difference  between  representative  a- values  for  the  peripheral  and 
central  parts  of  the  beam  is  unimportant,  but  it  must  be  re-examined  in 
tlie  drift  region  following  the  anode. 


392  THE   BELL   SYSTEM    TECHNICAL   JOURNAL,    MARCH    1956 

We  have  already  investigated  the  region  of  the  anode  hole  in  some 
detail  in  Section  3  and  have  found  it  worth  while  to  modify  the  ideal 
Davisson  expression  for  focal  length  of  an  equivalent  anode  lens.  In 
particular,  let  us  define  a  quantity  F  by 

F  =  focal  length  =  Fd/T  (19) 

where  Fd  is  the  Davisson  focal  length.  Thus  T  represents  a  corrective 
factor  to  be  applied  to  Fd  to  give  a  more  accurate  value  for  the  focal 
length.  In  so  far  as  any  thin  lens  is  capable  of  describing  the  effects  of 
diverging  fields  in  the  anode  region,  we  may  then  use  the  appropriate 
optical  formulas  to  transfer  our  knowledge  of  the  electron  trajectories 
(calculated  in  the  anode  region  as  outlined  above)  to  the  start  of  the  drift 
region.  In  particular, 

-f  (20) 

where  {dr/dz)i  and  {dr/dz)^  are  the  slopes  of  the  path  just  before  and 
just  after  the  lens,  and  r  is  the  distance  from  the  axis  to  the  point  where 
the  ideal  path  crosses  the  lens  plane. 

B.  The  Drift  Region 

Although  Te/a-  was  found  to  be  large  at  the  anode  plane  for  most  guns 
of  interest,  this  ratio  often  shrinks  to  1  or  less  at  an  axial  distance  of 
only  a  few  beam  diameters  from  the  lens.  Therefore,  the  assumption  that 
electron  trajectories  may  be  found  by  using  the  space  charge  forces 
which  would  exist  in  the  absence  of  thermal  velocities  of  emission  (i.e., 
forces  consistant  with  the  universal  beam  spread  curve)  may  lead  to  very 
appreciable  error.  For  example,  if  ecjual  normal  (Gaussian)  distributions 
of  points  about  a  central  point  are  superposed  so  that  the  central  points 
are  equally  dense  throughout  a  circle  of  radius  Te ,  and  if  the  standard  de- 
viation for  each  of  the  normal  distributions  is  cr  =  r^ ,  the  relative  density 
of  points  in  the  center  of  the  circle  is  only  about  39  per  cent  of  what  it 
would  be  Avith  a  <  (re/5). 

In  order  to  minimize  errors  of  this  type  we  have  modified  the  Hines- 
Cutler  treatment  of  the  drift  space  in  two  ways:  (1)  The  forces  influenc- 
ing the  trajectories  of  the  non- thermal  electrons  are  calculated  from  a 
progressive  estimation  of  the  actual  space  charge  configuration  as  modi- 
fied by  the  presence  of  thermal  electrons.  (2)  Some  account  is  taken  of 
the  fact  that,  as  the  space  charge  density  in  the  beam  becomes  less  uni- 
form as  a  function  of  radius,  the  spread  of  electrons  near  the  center  of 
the  beam  increases  more  rapidly  than  does  the  corresponding  spread 


BEAM    FORMATION    WITH    ELECTRON    GUNS  393 

farther  out.  Since  item  (1)  is  influenced  by  item  (2),  the  specific  as- 
sumptions involved  in  the  latter  case  will  be  treated  first. 

When  current  density  is  uniform  across  the  beam  and  its  cross  section 
changes  slowly  with  distance,  considerations  of  the  type  outlined  above 
for  the  gun  region  show  that  those  thermal  electrons  which  remain 
within  the  beam  will  continue  to  have  a  Gaussian  distribution  with  re- 
spect to  a  non-thermal  electron  emitted  from  the  same  cathode  point. 
When  current  density  is  not  uniform  over  the  cross  section,  we  would 
still  like  to  preserve  the  mathematical  simplicity  of  obtaining  the  current 
density  as  a  function  of  beam  radius  merely  by  superposing  Gaussian 
distributions  which  can  be  associated  with  each  non-thermal  electron. 
To  lessen  the  error  involved  in  this  simplified  approach,  we  will  arrive 
at  a  value  for  the  standard  deviation,  a  (which  specifies  the  Gaussian 
distribution),  in  a  rather  special  way.  In  particular,  a  at  any  axial  po- 
sition, z,  will  be  taken  as  the  radial  coordinate  of  an  electron  emitted 
from  the  center  of  the  cathode  with  a  transverse  velocity  of  emission 
given  by, 


ve  =  y- 


—  (21) 

m 


It  is  clear  from  (17)  that  for  such  an  electron,  r  =  o-  in  the  gun  region. 
From  (18),  the  fraction  of  the  electrons  from  a  common  point  on  the 
cathode  which  will  have  r  ^  a  in  the  gun  region  is 


2 


fraction  =    [   e'^'-'"-''^  d  ^=  I  -  e'"'  =  0.393  (22) 

If  re  denotes  the  radial  position  of  the  outermost  non-thermal  electron 
and  if  0-  >  /■,, ,  the  "a--electron"  will  be  moving  in  a  region  where  the 
space  charge  density  is  significantly  lower  than  at  the  axis.  We  could, 
of  course,  have  followed  the  path  of  an  electron  with  initial  velocity 
equal  to  say  0.1  or  10  times  that  given  in  (21)  and  called  the  correspond- 
nig  radius  O.lcr  or  lOo-.  The  reason  for  preferring  (21)  is  that  about  0.4 
or  nearly  half  of  the  thermal  electrons  emitted  from  a  common  cathode 
point  will  have  wandered  a  distance  less  than  a  from  the  path  of  a  non- 
thermal electron  emitted  from  the  same  cathode  point,  while  other 
thermal  electrons  will  ha\'e  wandered  farther  from  this  path;  conse- 
quently, the  current  density  in  the  region  of  the  o--electron  is  expected 
to  be  a  reasonable  average  on  which  beam  spreading  due  to  thermal 
\elocities  may  be  based.  With  this  understanding  of  how  a  is  to  be  cal- 
culated, we  can  proceed  to  the  calculation  of  non-thermal  electron 
trajectories  as  suggested  in  item  (1). 


394  THE    BELL   SYSTEM    TECHNICAL   JOURNAL,    MARCH    1956 

The  non-thermal  paths  remain  essentially  laminar,  and  with  r^  de- 
noting the  radial  coordinate  of  the  outermost  non-thermal  electron,  we 
will  make  little  error  in  assuming  that  the  current  density  of  non-ther- 
mal electrons  is  constant  for  r  <  Ve .  Consequently,  if  equal  numbers  of 
thermal  electrons  are  assumed  to  be  normally  distributed  about  the  cor- 
responding non-thermal  paths,  the  longitudinal  current  density  as  a 
function  of  radius  can  be  found  in  a  straightforward  way  by  using  (18). 
The  result  is 

J    ^  ^_(..,,..)   n"  R  ^-(«^/2.^)^^  frR\  ^  /R\  ^23) 

Jd  Jo        a  \a^/      \(t/ 

where  /o  is  the  zero  order  modified  Bessel  function  and  the  total  current 
is  Id  =  TTVe  Jd  '  Equation  (23)  was  integrated  to  give  a  plot  of  Jr/Jo 
versus  r/a,  with  re/a  as  a  parameter  and  is  given  as  Fig.  6  in  Reference 
6.  It  is  reproduced  here  as  Fig.  6.  Since  the  only  forces  acting  on  elec- 
trons in  the  drift  region  are  due  to  space  charge,  we  may  write  the  equa- 
tion of  motion  as 

where  Er  is  the  radial  electrical  field  acting  on  an  electron  with  radial 
coordinate  r.  Since  the  beam  is  long  and  narrow,  all  electric  lines  of  force 
may  be  considered  to  leave  the  beam  radially  so  that  Er  is  simpl}^  ob- 
tained from  Gauss'  law.  Equation  (24)  therefore  becomes 

-—   =  --^—  /    2irp  dr  =  -— ! —  /         ■  Iirr  dr 

dt^        zireor  Jo  Zireor  Jo    \/2t]V a. 


(25) 
2irenr       Jo 


27reor 

From  (23)  we  note  that  the  fraction  of  the  total  current  within  any 
radius  depends  only  on  fe/o-  and  j'/ct: 


:il 


dr 


^  /    J0')2irr  ar  /    xo     ,/o 

r  =  - =  H-)  f 

'-       r.J(r)2.rdr         ^''''°  (2«)  ' 

Jo 


■•r  I  a 

C 


'^dV^]^Fr-j- 

\(X    a  t 


\ 


BEAM  FORMATION  WITH  ELECTRON  GUNS 


395 


Fig.  6  —  Curves  showing  the  current  density  variation  with  radius  in  a  beam 
I  which  has  been  dispersed  by  thermal  velocities.  Here  r«  is  the  nominal  beam  radius, 
I    r  is  the  radius  variable,  and  <t  is  the  standard  deviation  defined  in  equation  17. 

A  family  of  curves  with  this  ratio,  Fr ,  as  parameter  has  been  reproduced 
:   from  the  Hines-Cutler  paper  and  appears  here  as  Fig.  7.  Using  this  no- 
tation, (25)  becomes 


dV  ^  Vr,/{2V.)  j^  Fr 
di^  27reo  r 


or 


d  r 
dz^ 


jn_         lo         Fr^         Fr 

27r€0  (27,7a)3/2    J.  J. 


(27) 


where  we  have  made  use  of  the  dc  electron  drift  velocity  to  make  dis- 
tance the  independent  variable  instead  of  time,  and  have  defined  a 
quantity  K  which  is  proportional  to  gun  perveance.  We  can  now  apply 
(27)  to  the  motion  of  both  the  outer  (edge)  non-thermal  electron  and 
the  cr-electron.  From  (26)  we  see  that  Fr,  and  Fg  depend  only  on  re/a] 


396  THE   BELL   SYSTEM   TECHNICAL  JOURNAL,   MARCH    1956 

12 


11 


10 


a 

LLI 
< 

u 

B    1 

z 

z 
o 

< 


o     ^ 


/^ 

,/;^ 

^ 

// 

P> 

^ 

Fr  = 

0.995/ 

^ 

^ 

/^ 

^ 

^ 

^ 

/^ 

/ 

rz 

^ 

/       >> 

/ 

^ 

z:^ 

^ 

:^ 

/ 

y. 

^ 

/y 

'A 

%: 

^ 

;^ 

^ 

Xy 

'^. 

^^ 

^^ 

^^ 

w 

i^ 

/^ 

oao^ 

^ 

1^ 

^- 

oo^ 

= 

^^ 

10 


re/0- 


Fig.  7  —  Curves  showing  the  fraction,  Fr  ,  of  the  total  beam  current  to  be  found 
within  any  given  radius  in  a  beam  dispersed  by  thermal  velocities  as  in  Fig.  6. 

consequently  the  continuous  solution  for  r^  and  r„  (=  a)  as  one  moves 
axially  along  the  drifting  beam  involves  the  simultaneous  solution  of  two 
equations : 


(fve 

d~a 
d^ 


KFr./re 
KFJa 


(28) 


BEAM    FORMATION    WITH    ELECTRON    GUNS 


397 


0.36 
0.32 
0.28 
0.24 

0.16 
0.12 
0.08 
0.04 
0 

\ 

\ 

1 

\ 

\ 

\ 

\ 

V 

V. 

--- 

— 

■ 

8 


10 


12 


14 


16 


I  Fig.  8  —  A  curve  showing  the  effect  of  a  quantity  related  to  the  space  charge 
•  force  (in  the  drift  region)  on  a  thermal  electron  with  standard  deviation  a.  (See 
'equation  28.) 


which  are  related  by  the  mutual  dependence  of  Fr^  and  Fa  on  re/a.  F„ 
and  Frjve  are  plotted  in  Figs.  8  and  9. 

We  may  summarize  the  treatment  of  the  drift  region,  then,  as  follows: 
1  (a)  The  input  values  of  r^  and  rgJ  at  the  entrance  to  the  anode  lens 
jare  obtained  from  the  Pierce  gun  parameters  r^  and  6,  while  the  value 
of  a  and  aJ  at  the  lens  entrance  can  be  obtained  as  mentioned  above 
by  integrating  (15)  from  the  cathode,  where  Mc  =  0  and  (dfx/dt)c  =  1, 
to  the  anode  plane.  (The  minus  subscripts  on  r'  and  a'  indicate  that 
these  slopes  are  being  evaluated  on  the  gun  side  of  the  lens;  a  plus  sub- 
script will  be  used  to  indicate  evaluation  on  the  drift  region  side  of  the 
lens.)  The  values  of  Ve  and  a  on  leaving  the  lens  will  of  course  be  their 
entrance  values  in  the  drift  region,  and  the  effect  of  the  lens  on  r/  and 
a'  is  simply  found  in  terms  of  the  anode  lens  correction  factor  T  by  use 
of  (20).  The  value  of  a  at  the  anode  can  be  obtained  from  (17)  if  n  is 
known  there.  In  this  regard,  (15)  can  be  integrated  once  to  give 


=  1_/M       dt 

"  "  r\dt)c{r,/r,y 


(29) 


398 


THE   BELL    SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


LL 


0.9 


0.8 


0.7 


0.6 


0.5 


0.4 


0.3 


0.2 


0.1 


. — 

— 

■-^ 

\ 

X 

^ 

\ 

I 

/ 

\ 

\    J 

\  / 

/ 

A 
7  \ 

/   \ 
/     \ 

\ 
\ 
\ 

\ 
\ 
\ 

\ 

f.,/(reA) 

\ 
\ 
\ 

s 

^, 

> 

"-. 

'"-.. 

^-*^^ 

■•—.^ 

1/ 
1 

0.38 


0.36 


0.34 


0.32 


0.30 


0.28 


0.26 


0.24 


0.22 


0.20 


0.18      1? 


0.1  6 


0.14 


0.12 


0.10 


0.08 


0.06 


0.04 


0.02 


6  7  8  9 


10 


11  12  13 


14 


Fig.  9  —  Showing  quantities  related  to  the  effect  of  the  space  charge  force  in 
the  drift  region  on  the  outermost  non-thermal  electron.  (See  equation  28.) 


i 


We  can  now  substitute  for  transit  time  in  terms  of  distance  and  Lang- 
muir's  well  known  potential  function/^  —a.  The  value  of  this  parameter, 
for  the  case  of  spherical  cathode-anode  geometry  in  which  we  are  in- 
terested, depends  only  on  the  ratio  fe/f  which  is  equal  to  Vc/rg .  (Because 
of  their  frerjuent  use  in  gun  design,  certain  functions  of  —a  are  included 
here  as  Table  I.)  In  terms  of  —a,  then,  the  potential  in  the  gun  region 


BEAM   FORMATION   WITH   ELECTRON   GUNS 


399 


Fable  I 


Table  of  Functions  of  —a  Often  Used  in  Electron 
Gun  Design 


fc/f 

(-«)2 

(-  a)V3 

(-  a)2/3 

difc/r) 

1.0 

0.0000 

0.0000 

0.0000 

0.0000 

1.025 

0.0006 

0.0074 

1.05 

0.0024 

0.0179 

0.134 

1.075 

0.0052 

0.0306 

0.173 

1.10 

0.0096 

0.0452 

0.212 

1.392 

0.590 

1.15 

0.0213 

0.0768 

0.277 

1.20 

0.0372 

0.1114 

0.334 

1.767 

0.716 

1.25 

0.0571 

0.1483 

0.385 

1.30 

0.0809 

0.1870 

0.432 

2.031 

0.790 

1.35 

0.1084 

0.2273 

0.476 

1.40 

0.1396 

0.2691 

0.519 

2.243 

0.874 

1.45 

0.1740 

0.3117 

0.558 

1.50 

0.2118 

0.3553 

0.596 

2.423 

0.886 

1.60 

0.2968 

0.4450 

0.667 

2.583 

0.915 

1.70 

0.394 

0.5374 

0.733 

2.725 

0.939 

1.80 

0.502 

0.6316 

0.795 

2.855 

0.954 

1.90 

0.621 

0.7279 

0.853 

2.975 

0.970 

2.00 

0.750 

0.8255 

0.908 

3.087 

0.982 

2.10 

0.888 

0.9239 

0.961 

3.192 

0.993 

2.20 

1.036 

1.024 

1.012 

3.292 

1.003 

2.30 

1.193 

1.125 

1.061 

3.388 

1.012 

2.40 

1.358 

1.226 

1.107 

3.481 

1.020 

2.50 

1.531 

1.328 

1.152 

3.570 

1.028 

2.60 

1.712 

1.431 

1.196 

3.655 

1.034 

2.70 

1.901 

1.535 

1.239 

3.738 

1.039 

2.80 

2.098 

1.639 

1.280 

3.817 

1.044 

2.90 

2.302 

1.743 

1.320 

3.894 

1.048 

3.00 

2.512 

1.848 

1.359 

3.968 

1.052 

3.1 

2.729 

1.953 

1.397 

4.040 

1.056 

3.2 

2.954 

2.059 

1.435 

4.111 

1.059 

3.3 

3.185 

2.164 

1.471 

4.180 

1.062 

3.4 

3.421 

2.270 

1.507 

4.247 

1.064 

3.5 

3.664 

2.376 

1.541 

4.315 

1.066 

3.6 

3.913 

2.483 

1.576 

4.377 

1.068 

3.7 

4.168 

2.590 

1.609 

4.441 

1.070 

3.8 

4.429 

2.697 

1.642 

4.501 

1.072 

3.9 

4.696 

2.804 

1.674 

4.563 

1.074 

4.0 

4.968 

2.912 

1.706 

4.621 

1.076 

400 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    1956 


may  be  written 


df  df      i-aaY'^ 


dt 


'V2nV  a/2;^  {-a) 


2/3 


(30) 
(31) 

(32) 


so  that  upon  substitution  from  (29)  and  (31),  (17)  becomes 
Fig.  5,  which  has  been  referred  to  above,  shows 

O-a      .   /2eVa 
'fcV     'kf- 

as  a  function  of  {fc/fa)  as  obtained  from  (32),  and  allows  o-„  to  be  de- 
termined easily.  Using  (20),  the  value  of  re+'  is  given  by 


/  Tea     , 


F 


-,.=  -^^_,.  =  ,/_g-l)     (33) 


where  dg  is  the  half-angle  of  the  cathode  (and  hence  the  initial  angle 
which  the  path  of  a  non-thermal  edge  electron  makes  with  the  axis). 
We  may  write  for  1/Fd 

1         V  fe  /d(-aY"\ 


Fo       4F       4(-aa)^/VV\rf(fc/r-)  7a 


(34) 


In  Fig.  10  we  plot  —falFr,  as  a  function  of  fjfa  for  easy  evaluation  of 
re+'  in  (33).  Taking  the  first  derivative  of  (32)  with  respect  to  ^,  we  ob- 
tain an  expression  for  aJ.  Using  this  in  conjunction  with  (20)  and  (34) 
we  find 


0-+     = 


Y  (r<^i  +  C2) 


I 


(35) 


where 


cira 


d{fc/f) 


/3 


and 


^-i/f.  ft -(-''"/ 


(-a)2/3_ 


! 


Ci  and  C2  are  plotted  as  functions  of  fc/fa  in  Fig.  11. 

(b)  After  choosing  a  specific  value  for  r  and  evaluating  K  =   rj/c/  . 


BEAM  FORMATION  WITH  ELECTRON  GUNS 


401 


Q 

LL 


lU 


I.O 

1.4 
1.2 
1.0 
0.8 
0.6 
0.4 
0.2 
0 

\ 

\ 

V 

\ 

\ 

~~~- 

■--- 

1.0        12  1.4        1.6         1.8        2.0       2.2       2.4       2.6        2.8       3.0       3.2       3.4        3.6       3.8        40 

rcAa 

Fig.  10  —  Curve  used  in  finding  ?•«+',  the  direction  of  a  nonthermal  edge  elec- 
tron as  it  enters  the  drift  region.  (See  equation  33.) 

(27r€o(277 Fa)  ''),  (28)  is  integrated  numerically  using  the  BTL  analog  com- 
puter to  obtain  a  and  r^  as  functions  of  axial  distance  along  the  beam, 
(c)  Knowing  a  and  Ve ,  other  beam  parameters  such  as  current  dis- 
tribution and  the  radius  of  the  circle  which  would  encompass  a  given 
percentage  of  the  total  current  can  be  found  from  Figs.  6  and  7. 


X 

tvi 

U 


20 

15 

10 

5 

0 

-5 

-10 

-15 

-20 

-25 

-30 


POLYNOMIAL   REPRESENTATION 
(ACCURATE    WITHIN    2°/o) 

-OR 

c,  & 

C2 

,'''' 

C,  =  4.13    fc/ra  +  2.67 

C2  =  0.635(r^/faf-13.56  rc/fa  +  19-33 

,  ,-' 

.' ' 

.-''' 

.'-' 

,'-' 

\ 

^^ 

-' 

^^-' 

< 

^v 

■^ 

X 

,.-' 

^** 

,^-' 

''H 

'^ 

"^ 

^ 

^ 

^^ 

\> 

^ 

20 
18 
16 
14 

12 

rO 
O 

10    X 

(J 

8 


2 

0 
10         1.2        1.4        1.6         1.8       2.0       2.2       2.4       2.6       2.8       3.0       3.2       3.4       3.6       3.8       4.0 

tc/fa 


Fig.  11  —  Curves  used  in  evaluating  o-+',  the  slope  of  the  trajectory  of  a  thermal 
electron  with  standard  deviation  a  as  it  enters  the  drift  region.  (See  equation  35.) 


402  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 

5.    NUMERICAL   DATA    FOR   ELECTRON   GUN   AND    BEAM   DESIGN 

A.  Choice  of  Variables 

Except  for  a  scaling  parameter,  the  electrical  characteristics  of  an 
ideal  Pierce  electron  gun  are  completely  determined  when  three  param- 
eters are  specified,  e.g.,  fc/fa  ,  perveance,  and  Va/T.  Also,  for  the  simp- 
lest case  r  is  equal  to  1  so  that  (since  K  depends  only  on  gun  perveance) 
in  this  case  no  additional  parameter  is  needed.  This  implies  that  nor- 
malized values  of  ?-/,  a,  a',  and  K  at  the  drift  side  of  the  anode  lens  are 
not  independent.  If,  however,  the  value  of  F  at  the  anode  lens  is  taken 
as  an  additional  variable,  four  parameters  plus  simple  scaling  are  re- 
quired before  complete  predictions  of  beam  characteristics  can  be  made. 
In  assembling  analog  computer  data  which  would  adequately  cover 
values  of  fc/fa  ,  perveance,  and  Va/T  which  are  likely  to  be  of  interest 
to  us  in  designing  future  guns,  we  chose  to  present  the  major  part  of 
our  data  with  T  fixed  at  1.1.  This  has  seemed  to  be  a  rather  typical  value 
for  r,  and  by  choosing  a  specific  value  we  decrease  the  total  number  of 
significant  variables  from  4  to  3.  (The  effect  of  variations  in  T  on  the 
minimum  radius  which  contains  95  per  cent  of  the  beam  is,  however, 
included  in  Fig.  16  for  particular  values  of  Va/T  and  perveance.)  Al- 
though the  boundary  conditions  for  our  mathematical  description  of  the 
beam  in  a  drift  space  are  simplest  when  expressed  in  terms  of  Vg ,  r/,  a 
and  ct',  we  have  attempted  to  make  the  results  more  usable  by  express- 
ing all  derived  parameters  in  terms  of  fc/fa  ,  s/Va/T,  and  the  perveance, 
P. 

B.  Tabular  Data 

The  rather  extensive  data  obtained  from  the  analog  computer  for  the 
r  =  1.1  case  and  for  practical  ranges  in  perveance,  Ve/T,  and  fc/fa 
are  summarized  in  Tables  IIA  to  E  where  the  parameters  r^  and  a  which 
specify  the  beam  cross  section  are  given  as  functions  of  axial  distance 
from  the  anode  plane.  Some  feeling  for  the  decrease  in  accuracy  to  be 
expected  as  the  distance  from  the  anode  plane  increases  can  be  obtained 
by  reference  to  Section  6B  where  experiment  and  theory  are  compared 
over  a  range  of  this  axial  distance  parameter. 

C.  Graphical  Data,  Including  Design  Charts  and  Beam  Profdes 

In  typical  cases,  the  designer  of  Pierce  electron  guns  is  much  more 
concerned  with  the  beam  radius  at  the  axial  position  where  it  is  smallest 
(and  in  the  axial  position  of  this  minimum)  than  he  is  in  the  general 


BEAM    FORMATION    WITH    ELECTRON    GUNS  403 


jspreadiug  of  the  beam  with  distance.  This  is  true  because,  in  microwave 
beam  tubes,  the  beam  from  a  magnetically  shielded  Pierce  gun  normally 
enters  a  strong  axial  magnetic  field  near  a  point  where  the  radius  is  a 
minimum,  so  that  magnetic  focusing  forces  largely  determine  the  beam's 
subsequent  behavior.  The  analog  computer  data  has  therefore  been  re- 
processed to  stress  the  dependence  of  the  beam's  minimum  diameter  and 
the  corresponding  axial  position  of  the  minimum  on  the  basic  design 
iparameters  fdfa  ,  perveance,  and  s/Va/T.  As  a  first  step  in  this  direc- 
tion, the  radius,  rgs ,  of  a  circle  which  includes  95  per  cent  of  the  beam 

:  I  current  is  obtained  as  a  function  of  axial  position  along  the  beam.  Such 
idata  are  shown  graphically  in  Fig.  12.  Finally,  the  curves  of  Fig.  12  are 

.  lused  in  conjunction  with  the  tabular  data  to  obtain  the  "Design  Curves" 
of  Fig.  13  where  all  of  the  pertinent  information  relating  to  the  beam 
at  its  minimum  diameter  is  presented. 

\D.  Example  of  Gun  Design  Using  Design  Charts 

Assume  that  we  desire  an  electron  gun  with  the  following  properties : 
anode  voltage  Va  =  1,080  volts,  cathode  current  Ip  =  7.1  ma,  and  mini- 
mum beam  diameter  2(r95)min  =  0.015  inches.  Let  us  further  assume  a 
cathode  temperature  T  =  1080°  Kelvin,  an  available  cathode  emission 
density  of  190  ma  per  square  cm,  and  an  anode  lens  correction  factor 
of  r  =  1.1.  From  these  data  we  find  -x/Va/T  =  1.0,  perveance  P  = 
0.2  X  10"^  amps/(volts)^''"  and  (r95)min/''c  =  0.174.  Reference  to  the  de- 
sign chart,  Fig.  13,  now  gives  us  the  proper  value  for  fc/fa  :  using  the 
upper  set  of  curves  in  the  column  for  y/Va/T  =1.0  we  note  the  point 
of  intersection  between  the  horizontal  line  for  {rgr^^i^/rc  =  0.174  and 
the  perveance  line  P  =  0.2,  and  read  the  value  of  fc/fa  (=  2.8)  as  the 
corresponding  abscissa.  The  convergence  angle  of  the  gun,  de ,  is  now 
simply  determined  fi'om  the  equation^^ 

de  =  cos-^  {\  -  t|^  X  10^)  (37) 

{Qe  is  found  to  be  13.7°  in  this  example)  and  the  potential  distribution 
in  the  region  of  the  cathode  can  be  obtained  from  (30). 

When  this  point  has  been  reached,  the  gun  design  is  complete  except 
for  the  shapes  of  the  beam  forming  electrode  and  the  anode,  which  are 
determined  with  the  aid  of  an  electrolytic  tank  in  the  usual  way.  The 
radius  of  the  anode  hole  which  will  give  a  specified  transmission  can  be 
found  by  obtaining  (re/a)a  through  the  use  of  Fig.  5,  and  then  choosing 
the  anode  radius  from  Fig.  7.  In  practical  cases  where  (rf/a)a  >  3.0, 


^' 


O 


O 

z 
o 

I-H 

e 
o 

w 

o 

O 

CO 

a 

Q 
O 
12; 

< 


O 
> 

m 

IS 
< 


Q  o 


P 

a, 

o 
O 


o 

< 

o 

m 


PQ 


"5 

d 
II 

> 

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I-H                                                1 

II                               ■! ' 

o 

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bl>? 

^H  (M  lO  O  C35  O  M        C^  t^  C^        GO 
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T-H  1-H  1—1  O  O  O  O  1"^  1"^  »~^  c^ 

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COiO^HOd-fiOiO         ^HCO-1"-f^H 

iooo^co»o^(MOt^coc;-t<co 

COIMtMi-HOOO^^CSMCO-* 

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GOiO^OOiOC^rt-tl^OO-liO 

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COOC0t^O'f--H--HCO'-HG0'*'Q0 

^Hi — ^r-HOOOOOO' — ^' — ^1 — 1 

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J  1  1  1  1  1 

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O(N^CD00O(M-*<£>Q0OC<) 

T— 1    1-H    T— t    T— 1    T-H    C^    C^l 

0(M^COGOOrt(N^cDO'*00 

t-Ht-Ht-Hi-H^05(MIM 

q 

b|=? 

O  IC  O  to  iO  lO  cr.  (M  C-j  t^  C^  C-j 
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BEAM   FORMATION   WITH   ELECTRON   GUNS  413 

we  find  less  than  1  per  cent  anode  interception  if 

anode  hole  radius  =  0.93  r^a  +  2o-a  (38) 

Additional  information  about  the  axial  position  of  (r95)min  and  the  cur- 
rent density  distribution  in  the  corresponding  transverse  plane  is  con- 
tained in  Fig.  13.  The  second  set  of  curves  in  the  \/Va/T  =  1  column 
gives  Zm\n/Tc  —  2.42  for  this  example,  so  that  we  would  predict 

Zmin  =  distance  from  anode  to  (r95)inin  =  0.104'' 

The  remaining  3''^  and  4*^^  sets  of  curves  in  the  ■\/Va/T  =  1  column 
allow  us  to  find  o-  and  re/a-  at  ^min  .  In  particular  we  obtain  a  =  0.0029" 
and  I'e/o  =  0.8,  and  use  Fig.  6  to  give  the  current  density  distribution  at 
2min  .*  Section  VI  contains  experimental  data  which  indicate  a  some- 
what larger  value  for  2m in  than  that  obtained  here.  However  the  pa- 
rameter of  greatest  importance,  (r95)niin  ,  is  predicted  with  embarrassing 
precision. 

For  those  cases  in  which  additional  information  is  required  about  the 
beam  shape  at  axial  points  other  than  ZnVin  ,  the  curves  of  Fig.  12  or  the 
data  of  Table  II  may  be  used. 

6.   COMPARISON    OF   THEORY  WITH   EXPERIMENT 

In  order  to  check  the  general  suitability  of  the  foregoing  theory  and 
the  usefulness  of  the  design  charts  obtained,  several  scaled-up  versions 
of  Pierce  type  electron  guns,  including  the  gun  described  in  Section  5D, 
were  assembled  and  placed  in  the  double-aperture  beam  analyzer  de- 
scribed in  Reference  7. 

A.  Measurement  of  Current  Densities  in  the  Beam 

Measurements  of  the  current  density  distributions  in  several  trans- 
verse planes  near  Smin  were  easily  obtained  with  the  aid  of  the  beam 
analyzer.  The  resulting  curve  of  relative  current  density  versus  radius 
at  the  experimental  2min  is  given  in  Fig.  14  for  the  gun  of  Section  52). 
(This  curve  is  further  discussed  in  Part  C  below.)  For  this  case,  as  well 
as  for  all  others,  special  precautions  were  taken  to  see  that  the  gun  was 
functioning  properly :  In  addition  to  careful  measurement  of  the  size  and 
position  of  all  gun  parts,  these  included  the  determination  that  the  dis- 
tribution of  transverse  velocities  at  the  center  of  the  beam  was  smooth 


*  When  j'c/o-  <  0.5,  the  current  density  distribution  depends  almost  entirely  on 
a,  and,  in  only  a  minor  way,  on  the  ratio  Te/a-  so  that  in  such  cases  this  ratio  need 
not  be  accurately  known. 


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THE    BELL   SYSTEM    TECHNICAL   JOURNAL,    MARCH    1956 


12 
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01  23456789 

RADIUS   IN     MILS 

Fig.  14  —  Current  density  distribution  in  a  transverse  plane  located  where  the 
95  per  cent  radius  is  a  minimum.  The  predicted  and  measured  curves  are  normal- 
ized to  contain  the  same  total  current.  (The  corresponding  prediction  from  the 
universal  beam  spread  curve  would  show  a  step  function  with  a  constant  relative 
current  density  of  64.2  for  r  <  1.2  mils  and  zero  beyond.)  The  gun  parameters  are 
given  in  Section  5D. 

and  generally  Gaussian  in  form,  thereby  indicating  uniform  cathode 
emission  and  proper  boundary  conditions  at  the  edge  of  the  beam  near 
the  cathode.  The  ejffect  of  positive  ions  on  the  beam  shape  was  in  every  I 
case  reduced  to  negligible  proportions,  either  by  using  special  pulse 
techniques,  or  by  applying  a  small  voltage  gradient  along  the  axis  of 
the  beam. 

B.  Comparison  of  the  Experiinentally  Measured  Spreading  of  a  Beam  with 
that  Predicted  Theoretically 

From  the  experimentally  obtained  plots  of  current  density  versus 
radius  at  several  axial  positions  along  the  beam,  we  have  obtained  at 
each  position  (by  integrating  to  find  the  total  current  within  any  radius) 
a  value  for  the  radius,  rgs ,  of  that  circle  which  encompasses  95  per  cent 
of  the  beam.  For  brevity,  we  call  the  resulting  plots  of  rgs  versus  axial 
distance,  "beam  profiles".  The  experimental  profile  for  the  giui  de- 
scribed in  Section  5D  is  shown  as  curve  A  in  Fig.  15(a).  Curve  B  shows 
the  profile  as  predicted  by  the  methods  of  this  paper  and  obtained  from 
Fig.  12.  Curve  C  is  the  corresponding  profile  which  one  obtains  by  the 
Hines-Cutler  method,   and  Curve  D  represents  Tq^  as  obtained  from  the 


BEAM  FORMATION  WITH  ELECTRON  GUNS 


417 


CO 


20 
18 
16 
14 
12 


t-       8 


2 
0 

I 

50 
45 
40 
35 


if)    30 


Z    25 

l?  20 

15 

10 


(a) 

GUN    PARAMETERS: 
fc/fa=2.8 

s 

/ 

^ 

\, 

(C)j 

/ 
1 

/ 

e  =  i3.7° 

VVa/T-i.o 

\ 

^> 

k 

/ 
/ 
/ 

[B]/ 

r 

rc  =  0.043" 

(A)  EXPERIMENT 

(B)  METHODS  OF 
THIS    PAPER 

(C)  HINES-CUTLER 
METHOD 

(D)  UNIVERSAL    BEAM 
SPREAD   CURVE 

\ 

^^ 

V 

/ 
/ 
/ 

/ 

/ 

\ 

N 

s. 

<; 

>^ 

^ 

'4 

/ 

^ 

<. 

\ 

\, 

"^ 

^>3e 

\ 

\, 

y 

/(D) 

\ 

\ 

y 

y 

"~~- 

^^ 

^ 

40 


80  120  160  200  240 

Z,  DISTANCE   FROM    IDEAL    ANODE    IN   MILS 


280 


320 


(b) 

i 

/(C) 

y 

GUN    PARAMETERS: 
f  c/fa  =  2.5 

1 
1 

1 

/ 

/ 

e  = 

8° 
1.0 

/ 

1 
* 

y 

^B) 

^/V, /T- 

\ 

x^ 

V      a/ 

rc  =  0.150" 

/ 

/ 
/ 
f 

y 

/^ 

V 

^ 

V 

^ 

^***^^ 

•■ 

• 

• 

} 

\ 

X 

X 

"^ 



,  -» 

-^ 

<^ 

— ■^ 
(A) 

\ 

^^ 

.^ 

y 

^-- 

^^ 

^ 

(D) 

100  200  300  400  500  600 

Z,  DISTANCE   FROM    IDEAL    ANODE    IN   MILS 


700 


800 


Fig.  15  —  Beam  profiles  (using  an  anode  lens  correction  of  r  =  1.1  and  the  gun 
parameters  indicated)  as  obtained  (A)  from  experiment,  (B)  bj^  the  methods  of  this 
paper,  (C)  Hines-Cutler  method,  (D)  by  use  of  the  universal  beam  spread  curve. 

universal  l^eam  spread  curve'"  (i.e.,  under  the  assumption  of  laminar 
flow  and  gradual  variations  of  beam  radius  with  distance) .  Note  that  in 
each  case  a  value  of  1.1  has  been  used  for  the  correction  factor,  r,  repre- 
senting the  excess  divergence  of  the  anode  lens.  The  agreement  in 
(/'95)min  as  obtaiucd  from  Curves  A  and  B  is  remarkably  good,  but  the 
axial  position  of  (r95)min  in  Curve  A  definitely  lies  beyond  the  correspond- 


418  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 

ing  inininumi  position  in  Curve  B.  Fortunately,  in  the  gun  design  stage, 
one  is  usually  more  concerned  with  the  value  of  (r95)min  than  with  its 
exact  axial  location.  The  principal  need  for  knowing  the  axial  location  of 
the  minimum  is  to  enable  the  axial  magnetic  field  to  build  up  suddenly 
in  this  neighborhood.  However,  since  this  field  is  normally  adjusted  ex- 
perimentally to  produce  best  focusing,  an  approximate  knowledge  of 
2m in  is  usually  adequate. 

In  Fig.  15b  we  show  a  similar  set  of  experimental  and  theoretical  beam 
profiles  for  another  gun.  The  relative  profiles  are  much  the  same  as  in 
Fig  15a,  and  all  of  several  other  guns  measured  yield  experimental 
points  similarly  situated  with  respect  to  curves  of  Type  B. 

C.  Comparison  of  Experimental  and  Theoretical  Current  Density  Dis- 
tributions where  the  Minimum  Beam  Diameter  is  Reached 

In  Fig.  14  we  have  plotted  the  current  density  distribution  we  would 
have  predicted  in  a  transverse  plane  at  ^min  for  the  example  introduced 
in  Section  5Z).  Here  the  experimental  and  theoretical  curves  are  nor- 
malized to  include  the  same  total  currents  in  their  respective  beams. 
The  noticeable  difference  in  predicted  and  measured  current  densities 
at  the  center  of  the  beam  does  not  appreciably  alter  the  properties  such 
a  beam  would  have  on  entering  a  magnetic  field  because  so  little  total 
current  is  actually  represented  by  this  central  peak. 

D.  Variation  of  Beam  Profile  with  T 

All  of  the  design  charts  have  been  based  on  a  value  of  T  =  1.1,  which 
is  typical  of  the  values  obtained  by  the  methods  of  Section  3.  When 
appreciably  different  values  of  F  are  appropriate,  we  can  get  some  feel- 
ing for  the  errors  involved,  in  using  curves  based  on  T  =  1.1,  by  refer- 
ence to  Fig.  16.  Here  we  show  beam  profiles  as  obtained  by  the  methods 
of  this  paper  for  three  values  of  F.  The  calculations  are  again  based  on 
the  gun  of  Section  5D,  and  a  value  of  just  over  1.1  for  F  gives  the  ex- 
perimentally obtained  value  for  (r95)min  . 

7.    SOME   ADDITIONAL   REMARKS    ON    GUN   DESIGN 

In  previous  sections  we  have  not  differentiated  between  the  voltage 
on  the  accelerating  anode  of  the  gun  and  the  final  beam  voltage.  It  is 
important,  howovei',  that  the  separate  functions  of  these  two  voltages 
be  kept  clearly  in  mind:  The  accelerating  anode  determines  the  total 
current  drawn  and  largely  controls  the  shaping  of  the  beam;  the  final 
beam  voltage  is,  on  the  other  hand,  chosen  to  give  maximum  interaction 
between  the  electron  beam  and  the  electromagnetic  waves  traveling 
along  the  slow  wave  circuit.  As  a  consequence  of  this  separation  of  func- , 


BEAM  FORMATION  WITH  ELECTRON  GUNS 


419 


0.006 


0.02 


0.18 


0.20 


0.22 


0.04  0.06  0.08  0.10  0.12  0.14  0.16 

Z,   DISTANCE    FROM    IDEAL   ANODE    IN    INCHES 

Fig.  16  —  Beam  profiles  as  obtained  by  the  methods  of  this  paper  for  the  gun 
parameters  given  in  Section  bD.  Curves  are  shown  for  three  values  of  the  anode 
lens  correction,  viz.  T  =  1.0,  1.1,  and  1.2. 

tions,  it  is  fouiicl  that  some  beams  which  are  difficult  or  impossible  to 
obtain  with  a  single  Pierce-gun  acceleration  to  final  beam  voltage  may 
be  obtained  more  easily  by  using  a  lower  voltage  on  the  gun  anode.  The 
acceleration  to  final  beam  voltage  is  then  accomplished  after  the  beam 
has  entered  a  region  of  axial  magnetic  field. 

Suppose,  for  example,  that  one  wishes  to  produce  a  2-ma,  4-kv  beam 
with  (rgs/rc)  =  0.25.  If  the  cathode  temperature  is  1000°K,  and  the  gun 
anode  is  placed  at  a  final  beam  voltage  of  4  kv,  we  have  \^Va/T  =  2 
and  P  =  0.008.  From  the  top  set  of  curves  under  \^Va/T  =  2  in  Fig. 
13,  we  find  (by  using  a  fairly  crude  extrapolation  from  the  curves  shown) 
that  a  ratio  of  fc/fa'^  3.5  is  required  to  produce  such  a  beam.  The  value 
of  {ve/o-)  at  Zmin  IS  therefore  less  than  about  0.2  so  that  there  is  little 
x'mblance  of  laminar  flow  here.  On  the  other  hand  we  might  choose 
r,  =  250  volts  so  that  a/fT^  =  0.5  and  P  =  0.51.  From  Fig.  13* 
we  than  obtain  fc/fa  =  2.6  and  (re/o-)min  =  0.8  for  the  same  ratio  of 
'■'joAc(=  0.25).  While  the  flow  could  still  hardly  be  called  laminar,  it  is 
(•(jnsiderably  more  ordered  than  in  the  preceding  case.  Here  we  have  in- 
cluded no  correction  for  the  (convergent)  lens  effect  associated  with  the 
post-anode  acceleration  to  the  final  beam  voltage,  F  =  4  kv. 

Calculations  of  the  Hines-Cutler  type  will  always  predict,  for  a  given 
set  of  gun  parameters  and  a  specified  anode  lens  correction,  a  minimum 
beam  size  which  is  larger  than  that  predicted  by  the  methods  of  this 
])aper.  Nevertheless,  in  many  cases  the  difference  between  the  minimum 
sizes  predicted  by  the  two  theories  is  negligible  so  long  as  the  same  anode 
lens  correction  is  used.  The  extent  to  which  the  two  theories  agree  ob- 


420  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    1956 

viously  depends  on  the  magnitude  of  Velo.  When  rel(T  as  calculated  by 
the  Hines-Cutler  method  (with  a  lens  correction  added)  remains  greater 
than  about  2  throughout  the  range  of  interest,  the  difference  between 
the  corresponding  values  obtained  for  rgs  will  be  only  a  few  per  cent. 
For  these  cases  where  rja  does  not  get  too  small,  the  principal  advan- 
tages of  this  paper  are  in  the  inclusion  of  a  correction  to  the  anode  lens 
formula  and  in  the  comparative  ease  with  which  design  parameters  may 
be  obtained.  In  other  cases  r^la  may  become  less  than  1,  and  the  theory 
presented  in  this  paper  has  extended  the  basic  Hines-Cutler  approach 
so  that  one  may  make  realistic  predictions  even  under  these  less  ideal 
conditions  where  the  departure  from  a  laminar-type  flow  is  quite  severe. 

ACKNOWLEDGMENT 

We  wish  to  thank  members  of  the  Mathematical  Department  at 
B.T.L.,  particularly  H.  T.  O'Neil  and  Mrs.  L.  R.  Lee,  for  their  help  in 
programming  the  problem  on  the  analog  computer  and  in  obtaining  the 
large  amount  of  computer  data  involved.  In  addition,  we  wish  to  thank 
J.  C.  Irwin  for  his  help  in  the  electrolytic  tank  work  and  both  Mr.  Irwin 
and  W.  A.  L.  Warne  for  their  work  on  the  beam  analyzer. 

REFERENCES 

1.  Pierce,  J.  R.,  Rectilinear  Flow  in  Beams,  J.  App.  Phys.,  11,  pp.  548-554,  Aug., 

1940. 

2.  Samuel,  A.  L.,  Some  Notes  on  the  Design  of  Electron  Guns,  Proc.  I.R.E.,  33, 

pp.  233-241,  April,  1945. 

3.  Field,  L.  M.,  High  Current  Electron  Guns,  Rev.  Mod.  Phys.,  18,  pp.  353-361, 

July,  1946. 

4.  Davisson,  C.  J.,  and  Calbick,  C.  J.,  Electron  Lenses,  Phys.  Rev.,  42,  p.  580, 

Nov.,  1932. 

5.  Helm,  R.,  Spangenburg,  K.,  and  Field,  L.  M.,  Cathode-Design  Procedure  for 

Electron  Beam  Tubes,  Elec.  Coram.,  24,  pp.  101-107,  March,  1947. 

6.  Cutler,  C.  C,  and  Hines,  M.  E.,  Thermal  Velocity  Effects  in  Electron  Guns, 

Proc.  I.R.E.,  43,  pp.  307-314,  March,  1955. 

7.  Cutler,  C.  C,  and  Saloom,  J.  A.,  Pin-hole  Camera  Investigation  of  Electron 

Beams,  Proc.  I.R.E.,  43,  pp.  299-306,  March,  1955. 

8.  Hines,  M.  E.,  Manuscript  in  preparation. 

9.  Private  communication. 

10.  See  for  example,  Zworykin,  V.  K.,  et  al..  Electron  Optics  and  the  Electron 

Microscope,  Chapter  13,  Wiley  and  Sons,  1945,  or  Klemperer,  O.,  Electron 
Optics,  Chapter  4,  Cambridge  Univ.  Press,  1953. 

11.  Brown,  K.  L.,  and  Siisskind,  C.,  The  Effect  of  the  Anode  Aperature  on  Po- 

tential Distribution  in  a  "Pierce"  Electron  Gun,  Proc.  I.R.E.,  42,  p.  598, 
March,  1954. 

12.  See,  for  example,  Pierce,  J.  R.,  Theory  and  Design  of  Electron  Beams,  p.  147, 

Van  Nostrand  Co.,  1949. 

13.  See  Reference  6,  p.  5. 

14.  Langmuir,  I.  L.,  and  Blodgett,  K.,  Currents  Limited  by  Space  Charge  Be- 

tween Concentric  Spheres,  Phys.  Rev.,  24,  p.  53,  July,  1924. 

15.  See  Reference  12,  p.  177. 

16.  See  Reference  12,  Chap.  X. 


Theories  for  Toll  Traffic  Engineering  in 

the  U.S.A.* 

By  ROGER  I.  WILKINSON 

(Manuscript  received  June  2,  1955) 

Present  toll  trunk  traffic  engineering  practices  in  the  United  States  are 
reviewed,  and  various  congestion  formulas  compared  with  data  obtained  on 
long  distance  traffic.  Customer  habits  upon  meeting  busy  channels  are  noted 
and  a  theory  developed  describing  the  probable  result  of  permitting  subscribers 
to  have  direct  dialing  access  to  high  delay  toll  trunk  groups. 

Continent-wide  automatic  alternate  routing  plans  are  described  briefly, 
in  which  near  no-delay  service  will  permit  direct  customer  dialing.  The 
presence  of  non-random  overflow  traffic  from  high  usage  groups  co7nplicates 
the  estimation  of  correct  quantities  of  alternate  paths.  Present  methods  of 
solving  graded  multiple  problems  are  reviewed  and  found  unadaptable  to  the 
variety  of  trunking  arrangements  occurring  in  the  toll  plan. 

Evidence  is  given  that  the  principal  fluctuation  characteristics  of  overflow- 
type  of  non-random  traffic  are  described  by  their  mean  and  variance.  An 
approximate  probability  distribution  of  simultaneous  calls  for  this  kind  of 
non-random  traffic  is  developed,  and  found  to  agree  satisfactorily  with  theo- 
retical overflow  distributions  and  those  seen  in  traffic  simidations. 

A  method  is  devised  using  ^^ equivalent  random''^  traffic,  which  has  good 
loss  predictive  ability  under  the  "lost  calls  cleared"  assumption,  for  a  diverse 
field  of  alternate  route  trunking  arrangements.  Loss  comparisons  are  made 
with  traffic  simulation  residts  and  with  observations  in  exchanges. 

Working  curves  are  presented  by  which  midti-alternate  route  trunking 
systems  can  be  laid  out  to  meet  economic  and  grade  of  service  criteria.  Exam- 
ples of  their  application  are  given. 

Table  of  Contents 

1 .  Introduction 422 

2.  Present  Toll  Traffic  Engineering  Practice 423 


*  Presented  at  the  First  International  Congress  on  the  Application  of  the 
Theory  of  Probability  in  Telephone  Engineering  and  Administration,  Copen- 
hagen, June  21,  1955. 

421 


422  THE   BELL  SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 

3.  Customers  Dialing  on  Groups  with  Considerable  Delay 431 

3.1.  Comparison  of  Some  Formulas  for  Estimating  Customers'  NC  Service 

on  Congested  Groups 434 

4.  Service  Requirements  for  Direct  Distance  Dialing  by  Customers 436 

5.  Economics  of  Toll  Alternate  Routing 437 

6.  New  Problems  in  the  Engineering  and  Administration  of  Intertoll  Groups 
Resulting  from  Alternate  Routing 441 

7.  Load-Service  Relationships  in  Alternate  Route  Systems 442 

7.1.  The  "Peaked"  Character  of  Overflow  Traffic 443 

7.2.  Approximate  Description  of  the  Character  of  Overflow  Traffic 446 

7.2.1.  A  Probability  Distribution  for  Overflow  Traffic 452 

7.2.2.  A  Probability  Distribution  for  Combined  Overflow  Traffic  Loads  457 

7.3.  Equivalent  Random  Theory  for  Prediction  of  Amount  of  Traffic  Over- 
flowing a  Single  Stage  Alternate  Route,  and  Its  Character,  with  Lost 

Calls  Cleared 461 

7.3.L  Throwdown  Comparisons  with  Equivalent  Random  Theory  on 

Simple    Alternate    Routing    Arrangements    with    Lost     Calls 

Cleared 468 

7.3.2.  Comparison  of  Equivalent  Random  Theory  with  Field  Results 

on  Simple  Alternate  Routing  Arrangements 470 

7.4.  Prediction  of  Traffic  Passing  Through  a  Multi-Stage  Alternate  Route 

Network 475 

7.4.1.  Correlation  of  Loss  with  Peakedness  of  Components  of  Non- 
Random  Offered  Traffic 481 

7.5.  Expected  Loss  on  First  Routed  Traffic  Offered  to  Final  Route 482 

7.6.  Load  on  Each  Trunk,  Particularly  the  Last  Trunk,  in  a  Non-Slipped 
Alternate  Route 486 

8.  Practical  Methods  for  Alternate  Route  Engineering 487 

8.1.  Determination  of  Final  Group  Size  with  First  Routed  Traffic  Offered 
Directly  to  Final  Group 490 

8.2.  Provision  of  Trunks  Individual  to  First  Routed  Traffic  to  Equalize 
Service 491 

8.3.  Area  in  Which  Significant  Savings  in  Final  Route  Trunks  are  Real- 
ized by  Allowing  for  the  Preferred  Service  Given  a  First  Routed 
Traffic  Parcel 494 

8.4.  Character  of  Traffic  Carried  on  Non-Final  Routes 495 

8.5.  Solution  of  a  Typical  Toll  Multi-Alternate  Route  Trunking  Arrange- 
ment :  Bloomsburg,  Pa 500 

9.  Conclusion 505 

Acknowledgements 506 

References 506 

Abridged  Bibliography  of  Articles  on  Toll  Alternate  Routing 507 

Appendix  I:  Derivation  of  Moments  of  Overflow  Traffic 507 

Appendix  II:  Character  of  Overflow  when  Non-Random  Traffic  is  Offered 

to  a  group  of  Trunks 511 


1.    INTRODUCTION 

It  has  long  been  the  stated  aim  of  the  Bell  System  to  make  it  easily 
and  economically  possible  for  any  telephone  customer  in  the  United 
States  to  reach  any  other  telephone  in  the  world.  The  principal  effort 
in  this  direction  by  the  American  Telephone  and  Telegraph  Company 
and  its  associated  operating  companies  is,  of  course,  confined  to  inter- 
connecting the  telephones  in  the  United  States,  and  to  providing  com- 
munication channels  between  North  America  and  the  other  countries  of 
the  world.  Since  the  United  States  is  some  1500  miles  from  north  to 
fSOuth  and  3000  miles  from  east  to  west,  to  realize  even  the  aim  of  fast 


THEORIES   FOR   TOLL   TRAFFIC   ENGINEERING    IN   THE   U.    S.    A.      423 

and  economical  service  between  customers  is  a  problem  of  great  magni- 
tude; it  has  engaged  our  planning  engineers  for  many  years. 

There  are  now  52  million  telephones  in  the  United  States,  over  80  per 
cent  of  which  are  equipped  with  dials.  Until  quite  recently  most  telephone 
users  were  limited  in  their  direct  dialing  to  the  local  or  immediately  sur- 
rounding areas  and  long  distance  operators  were  obliged  to  build  up  a 
circuit  with  the  aid  of  a  "through"  operator  at  each  switching  point. 

Both  speed  and  economy  dictated  the  automatic  build-up  of  long  toll 
circuits  without  the  intervention  of  more  than  the  originating  toll  oper- 
ator. The  development  of  the  No.  4-type  toll  crossbar  switching  system 
with  its  ability  to  accept,  translate,  and  pass  on  the  necessary  digits  (or 
lujuivalent  information)  to  the  distant  office  made  this  method  of  oper- 
ation possible  and  feasible.  It  was  introduced  during  World  War  II,  and 
now  by  means  of  it  and  allied  equipment,  55  per  cent  of  all  long  distance 
calls  (over  25  miles)  are  completed  by  the  originating  operator. 

As  more  elaborate  switching  and  charge-recording  arrangements  were 
developed,  particularly  in  metropolitan  areas,  the  distances  which  cus- 
tomers themselves  might  dial  measurably  increased.  This  expansion  of 
the  local  dialing  area  was  found  to  be  both  economical  and  pleasing  to 
the  users.  It  was  then  not  too  great  an  effort  to  visualize  customers 
dialing  to  all  other  telephones  in  the  United  States  and  neighboring 
countries,  and  perhaps  ultimately  across  the  sea. 

The  physical  accomplishment  of  nationwide  direct  distance  dialing 
which  is  now  gradually  being  introduced  has  involved,  as  may  well  be 
imagined,  an  immense  amount  of  advance  study  and  fundamental  plan- 
ning. Adequate  transmission  and  signalling  with  up  to  eight  intertoll 
trunks  in  tandem,  a  nationwide  uniform  numbering  plan  simple  enough 
to  be  used  accurately  and  easily  by  the  ordinary  telephone  caller,  pro- 
^  ision  for  automatic  recording  of  who  called  whom  and  how  long  he 
talked,  with  subsequent  automatic  message  accounting,  are  a  few  of 
man}^  problems  which  have  required  solution.  How  they  are  being  met  is 
a  romantic  story  beyond  the  scope  of  the  present  paper.  The  references 
given  in  the  bibliography  at  the  end  contain  much  of  the  history  as  well 
as  the  plans  for  the  future.  • 

2.    PRESENT   TOLL   TRAFFIC    ENGINEERING   PRACTICE 

There  are  today  approximately  116,000  intertoll  trunks  (over  25  miles 
in  length)  in  the  Bell  System,  apportioned  among  some  13,000  trunk 
groups.  A  small  segment  of  the  2,600  toll  centers  which  they  interconnect 
is  shown  in  Fig.  1.  Most  of  these  intertoll  groups  are  presently  traffic 
engineered  to  operate  according  to  one  of  several  so-called  T-schedules: 
T-8,  T-15,  T-30,  T-60,  or  T-120.  The  number  following  T  (T  for  Toll)  is 


424  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    1956 


KEY 

O      TOLL    CENTERS 

INTERTOLL    TRUNK    GROUPS 


Fig.  1  —  Principal  intertoll  trunk  groups  in  Minnesota  and  Wisconsin. 


THEORIES    FOR   TOLL   TRAFFIC    ENGINEERING   IN   THE   U.    S.    A.      425 


4       5      6     7   8  9  10 
NUMBER    OF  TRUNKS 


30       40    50 


Fig.  2  —  Permitted  intertoU  trunk  occupancy  for  a  6.5-minute  usage  time 
per  message. 

the  expected,  or  average,  delay  in  seconds  for  calls  to  obtain  an  idle 
trunk  in  that  group  during  the  average  Busy  Season  Busy  Hour.  In  1954 
the  system  "average  trunk  speed"  was  approximately  30  seconds,  re- 
sulting from  operating  the  majority  of  the  groups  at  a  busy-hour  trunk- 
ling  efficiency  of  75  to  85  per  cent  in  the  busy  season. 

The  T-engineering  tables  show  permissible  call  minutes  of  use  for  a 


426  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    1956 

wide  range  of  group  sizes,  and  several  selections  of  message  holding 
times.  They  were  constructed  following  summarization  of  many  obser- 
vations of  load  and  resultant  average  delays  on  ringdown  (non-dial) 
intertoll  trunks.^  Fig.  2  shows  the  permissible  occupancy  (efficiency)  of 
various  trunk  group  sizes  for  6.5  minutes  of  use  per  message,  for  a  va- 
riety of  T-schedules.  It  is  perhaps  of  somfe  interest  that  the  best  fitting 
curves  relating  average  delay  and  load  were  found  to  be  the  well-known 
Pollaczek-Crommelin  delay  curves  for  constant  holding  time  —  this  in 
spite  of  the  fact  that  the  circuit  holding  times  were  far  indeed  from 
having  a  constant  value. 

A  second,  and  probably  not  uncorrected,  observation  was  that  the 
per  cent  "No-Circuit"  (NC)  reported  on  the  operators'  tickets  showed 
consistently  lower  values  than  were  measured  on  group-busy  timing  de- 
vices. Although  not  thoroughly  documented,  this  disparity  has  generally 
been  attributed  to  the  reluctance  of  an  operator  to  admit  immediately 
the  presence  of  an  NC  condition.  She  exhibits  a  certain  tolerance  (very 
difficult  to  measure)  before  actually  recording  a  delay  which  would 
recjuire  her  to  adopt  a  prescribed  procedure  for  the  subsequent  handling 
of  the  call.*  There  are  then  two  measures  of  the  No-Circuit  condition 
which  are  of  some  interest,  the  "NC  encountered"  by  operators,  and  the 
"NC  existing"  as  measured  by  timing  devices. 

It  has  long  been  observed  that  the  distribution  of  numbers  n  of  simul- 
taneous calls  found  on  T-engineered  ringdown  intertoll  groups  is  in  re- 
markable agreement  with  the  individual  probability  terms  of  the  Erlang 
"lost  calls"  formula, 

f  n  — a ' 

a   e 


fin)  =  ^-^^  (1) 

e 


E- 


n=o     n! 

where  c  =  number  of  paths  in  the  group, 

a'  =  an    enhanced    average    load    submitted    such    that 
a'[l  —  Ei^c(a')]  =  L,  the  actual  load  carried,  and 
Ei^cid')  =  fie)  =  Erlang  loss  probability  (commonly  called  Er- 
lang B  in  America). 
An  example  of  the  agreement  of  observations  with  (1)  is  shown  in  Fig. 
3,  where  the  results  of  switch  counts  made  some  years  ago  on  many 
ringdown  circuit  groups  of  size  3  are  summarized.  A  wide  range  of  "sub- 


*  Upon  finding  No-Circuit,  an  operator  is  instructed  to  try  again  in  30  seconds 
and  GO  seconds  (before  giving  an  NC  report  to  the  customer),  followed  by  addi- 
tional attempts  5  minutes  and  10  minutes  later  if  necessary. 


THEORIES    FOR   TOLL   TRAFFIC    ENGINEERING   IN   THE   U.    S.    A.      427 


0.10  0.2  0.5  1.0  2 

AVERAGE  "submitted"  LOAD    IN    ERLANIGS 


Fig.  3  —  Distributions  of  simultaneous  calls  on  three-trunk  toll  groups  at 
.\lbany  and  Buffalo. 

I  nit  ted"  loads  a'  to  produce  the  observed  carried  loads  is  required.  On 
Fig.  4  are  shown  the  corresponding  comparisons  of  theory  and  obser- 
vations for  the  proportions  of  time  all  paths  are  busy  ("NC  Existing") 
for  2-,  4-,  5-,  7-,  and  9-circuit  groups.  Good  agreement  has  also  been  ob- 
served for  circuit  groups  up  to  20  trunks.  This  has  been  found  to  be  a 
stable  relationship,  in  spite  of  the  considerable  variation  in  the  actual 
practices  in  ringdown  operation  on  the  resubmission  of  delayed  calls. 
Since  the  estimation  of  traffic  loads  and  the  subsequent  administration 
of  ringdown  toll  trunks  has  been  performed  principally  by  means  of 
Group  Busy  Timers  (which  cumulate  the  duration  of  NC  time),  the 
Erlang  relationship  just  described  has  been  of  great  importance. 


428 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


With  the  recent  rapid  increase  in  operator  dialed  intertoll  groups,  it 
might  be  expected  that  the  above  discrepancy  between  "  %  NC  encoun- 
tered" and  "%  NC  existing"  would  disappear  —  for  an  operator  now 
initiates  each  call  unaware  of  the  momentary  state  of  the  load  on  any 
particular  intertoll  group.  By  the  use  of  peg  count  meters  (which  count 
calls  offered)  and  overflow  call  counters,  this  change  has  in  fact  been 
observed  to  occiu'.  ]\Ioreo^'er,  since  the  initial  re-trial  intervals  are  com- 
monly fairly  short  (30  seconds)  subsequent  attempts  tend  to  find  some 
of  the  previous  congestion  still  existing,  so  that  the  ratio  of  overflow  to 
peg  count  readings  now  exceeds  slightly  the  "%  NC  existing."  This 
situation  is  illustrated  in  Fig.  5,  which  shows  data  taken  on  an  operator- 


1.0 


AVERAGE    SUBMITTED    LOAD 


Fig.  4  —  Observed  proportions  of  time  all  trunks  were  busy  on  Albany  and 
Buffalo  groups  of  2,  4,  5,  7,  and  9  trunks, 


THEORIES   FOR   TOLL   TRAFFIC   ENGINEERING   IN   THE   U.    S.    A.      429 


u 

z 

o 

z 
I- 

UJ 
HI 

5 

ul 
_i 
_i 
< 
o 

U- 

o 

z 
o 

(- 
cc 
o 
a. 
o 
tr 
a. 


0.001 


12  14 

L  =  LOAD   CARRIED    IN    ERLANGS 


18 


Fig.  5  —  Comparison  of  NC  data  on  a  16-trunk  T-engineered  toll  group  with 
various  load  versus  NC  theories. 


dialed  T-engineered  group  of  16  trunks  between  Newark,  N.  J.,  and 
Akron,  Ohio.  Curve  A  shows  the  empirically  determined  "NC  encoun- 
tered" relationship  described  above  for  ringdown  operation;  Curve  B 
gives  the  corresponding  theoretical  "NC  existing"  values.  Lines  C  and  D 
give  the  operator-dialing  results,  for  morning  and  afternoon  busy  hours. 
The  observed  points  are  now  seen  generally  to  be  significantly  above 
Curve  B.* 

At  the  same  time  as  this  change  in  the  "NC  encountered"  was  occur- 
ring, due  to  the  introduction  of  operator  toll  dialing,  there  seems  to  have 
l)een  little  disturbance  to  the  traditional  relationship  between  load 

*  The  observed  point  at  11  erlangs  which  is  clearly  far  out  of  agreement  with 
the  remainder  of  the  data  was  produced  by  a  combination  of  high-trend  hours 
and  an  hour  in  which  an  operator  apparently  made  many  re-t^rials  in  rapid  suc- 
cession. 


430 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    1956 


u. 

10 

z 

o 

o 

5 

o 
m 

rvj 

ti 

_r 
< 

(- 

z 

LU 

z 
o 

z 
o 

i 

tr 

UJ 

I/) 

§ 

o 

«---   LIMIT  OF 
OBSERVED 
DATA 

i 

[ 

oiT 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 
/ 

/ 

y 

/ 

/ 

/' 

^•^ 

/^ 

^ 

^ 

««- 

^ 

^ 

Tt^^ 

^•^^ 

^ 

s:;^ 

8 


If) 


o 

0> 


o 

(0 


(O 


o 


If) 


o 


in 


o 


o 


in 
tvj 


o       in 


SBIONII^    1-    a3AO  SidlAjaiiV   dO    iN3D«3d 


THEORIES    FOR   TOLL   TRAFFIC   ENGINEERING   IN   THE   U.    S.   A.      431 

carried  and  "  %  NC  existing."  C.  J.  Truitt  of  the  A.T.  &  T.  Co.  studied 
i  a  number  of  operator-dialed  T-engineered  groups  at  Newark,  New  Jersey, 
in  1954  with  a  traffic  usage  recorder  (TUR)  and  group-busy  timers,  and 
found  the  relationship  of  equation  (1)  still  good.  (This  analysis  has  not 
been  published.) 

A  study  by  Dr.  L.  Kosten  has  provided  an  estimate  of  the  probability 
that  when  an  NC  condition  has  been  found,  it  will  also  appear  at  a  time 
T  later."  When  this  modification  is  made,  the  expected  load-versus-NC 
relationship  is  shown  by  Curve  E  on  Fig.  5.  (The  re-trial  time  here  was 
taken  as  the  operators'  nominal  30  seconds;  with  150-second  circuit-use 
time  the  return  is  0.2  holding  time.)  The  observed  NC's  are  seen  to  lie 
slightly  above  the  E-curve.  This  could  be  explained  either  on  the  basis 
that  Kosten's  analysis  is  a  lower  limit,  or  that  the  operators  did  not 
strictly  observe  the  30-second  return  schedule,  or,  more  probably,  a 
combination  of  both. 

3.    CUSTOMERS  DIALING  ON  GROUPS  WITH  CONSIDERABLE  DELAY 

It  is  not  to  be  expected  that  customers  could  generally  be  persuaded  to 
wait  a  designated  constant  or  minimum  re-trial  time  on  their  calls  which 
meet  the  NC  condition.  Little  actual  experience  has  been  accumulated 
on  customers  dialing  long  distance  calls  on  high-delay  circuits.  However, 
it  is  plausible  that  they  would  follow  the  re-trial  time  distributions  of 
customers  making  local  calls,  who  encounter  paths-busy  or  line-busy 
signals  (between  which  they  apparently  do  not  usually  distinguish). 
Some  information  on  re-trial  times  was  assembled  in  1944  by  C.  Clos  by 
observing  the  action  of  customers  who  received  the  busy  signal  on  1,100 
local  calls  in  the  City  of  New  York.  As  seen  in  Fig.  6,  the  return  times, 
after  meeting  "busy,"  exhibit  a  marked  tendency  toward  the  exponential 
distribution,  after  allowance  for  a  minimum  interval  required  for  re- 
dialing. 

An  exponential  distribution  with  average  of  250  seconds  has  been 
I  fitted  by  eye  on  Fig.  6,  to  the  earlier  ■ —  and  more  critical  —  customer  re- 
turn times.  This  may  seem  an  unexpectedly  long  wait  in  the  light  of  indi- 
vidual experience;  however  it  is  probably  a  fair  estimate,  especially 
since,  following  the  collection  of  the  above  data,  it  has  become  common 
practice  for  American  operating  companies  in  their  instructional  lit- 
erature to  advise  customers  receiving  the  busy  signal  to  "hang  up,  wait 
a  few  minutes,  and  try  again." 

The  mathematical  representation  of  the  situation  assuming  exponen- 
I  tial  return  times  is  easily  formulated.  Let  there  be  .r  actual  trunks,  and 


432  THE   BELL   SYSTEM   TECHNICAL  JOURNAL,  MARCH    1956 

imagine  y  waiting  positions,  whore  y  is  so  large  that  few  calls  are  re- 
jected.* Assume  that  the  offered  load  is  a  erlangs,  and  that  the  calls  have 
exponential  conversation  holding  times  of  unit  average  duration.  Finally  \ 
let  the  average  return  time  for  calls  which  have  advanced  to  the  waiting  > 
positions,  be  1/s  times  that  of  the  unit  conversation  time.  The  statistical  j 
equilibrium  equation  can  then  be  written  for  the  probability  j\m,  n)  (j 
that  m  calls  are  in  progress  on  the  x  trunks  and  n  calls  are  waiting  on 
the  y  storage  positions:  ■ 

/(w,  n)  =  aj{m  —  1,  n)  dt  +  s(w  +  l)/(m  —  1,  n  +  1)  dt  ''■) 

+  (m  +  \)J{m  +  1,  n)  dt  +  a/(.r,  n  -  1)  dH^  (2) 

+  [1  -  (a***  +  sn**)  dt  -  m  dt]f(m,  n)  ^ 

where  0  ^  m  ^  .-r,  0  ^  w  ^  //,  and  the  special  limiting  situations  are 
recognized  by: 

■*  Include  term  only  when  m  —  x 

**■  Omit  sn  when  m  =  x 

***  Omit  a  when  m  =  x  and  n  =  y 

Equation  (2)  reduces  to 

(a***  +  snifif  +  m)f{m;  n)  =  af{m  —  1,  n)  1 

+  s(n  +  l)/(m  -  1,  w  +  1)  (3) 

+  (m  +  l)/(w  +  1,  n)  +  af(x,  n  -  !)•, 

Solution  of  (3)  is  most  easily  effected  for  moderate  values  of  x  and  y 
by  first  setting  f(x,  ?/)  =   1 .000000  and  solving  for  all  other  /(/?? ,  ?? )  in 

X        y 

terms  of /(o:,  ?/).  Normahzing  through  zl  11f(m,  n)   =   1.0,  then  gives 

m=0   n=0 

the  entire  f(m,  n)  array. 

The  proportion  of  time  "NC  exists,"  will,  of  course  be 

Z  Six,  n)  (4) 

n=0 

and  the  load  carried  is 

L  =  Xl  X  wi/(m,  n)  (5) 

The  proportion  of  call  attempts  meeting  NC,  including  all  re-trials 


*  The  quant  itjr  y  can  also  be  chosen  so  that  some  calls  are  rejected,  thus  roughly 
describing  those  calls  abandoned  after  the  first  attempt. 


THEORIES   FOR  TOLL  TRAFFIC    ENGINEERING   IN   THE   U.   S.    A,      433 


will  be 


W{x,  a,  s)  = 


Expected  overflow  calls  per  unit  time 
Expected  calls  offered  per  unit  time 

Z  (a  +  sn)jXx,  n)  -    ,      ./       ^  ^^^ 

sn  -\-  af{x,  y) 


n=0 


X         y 


S  2  («  +  sn)f(m,  n) 


a  -{-  sn 


m=0    71=0 


X         y 


in  which  n  =   ^  2^  nf(7n,  n).  And  when  y  is  chosen  so  large  that/(.r,  y) 

7H  =  0     71=0 

is  negligible,  as  we  shall  use  it  here, 


L  =  a 


W(x,  a,  s)  = 


sn 


a  -\-  sn 


(5') 
(6') 


1^       0.5 

< 

"^O    0.4 
ilZ 

Oo 
ZZ    0.3 

Ol- 

pllJ 

o5   0.2 

Q. 

o 

?       0.1 


6  TRUNKS 


/        //      APOISSON 
'         ^1  P(C,L) 


5=0.6 


2  4  6  8 

L=LOAD  CARRIED    IN   ERLANGS 


APOISSON 
P(C,L) 


fly    >^- 

f  I6j      _, 


8  10  12  14 

L  =  LOAD  CARRIED  IN   ERLANGS 


Fig.  7  — ■  Comparison  of  trunking  formulas. 


434  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    1956 


I 


This  formula  provides  a  means  for  estimating  the  grade  of  service 
which  customers  might  he  expected  to  receive  if  asked  to  dial  their  calls 
over  moderate-delay  or  high-delay  trunk  groups.  For  a  circuit  use  length 
of  150  seconds,  and  an  average  return  time  of  250  seconds  (as  on  Fig.  6), 
both  exponential,  the  load-versus-proportion-NC  curves  for  6  and  IG 
trunks  are  given  as  curves  (3)  on  Fig.  7.  For  example  with  an  offered 
(=  carried)  load  of  a  =  4.15  erlangs  on  6  trunks  we  should  expect  to  find 
27.5  per  cent  of  the  total  attempts  resulting  in  failure. 

For  comparison  with  a  fixed  return  time  of  NC-calls,  the  IF-formula 
curves  for  exponential  returns  of  30  seconds  (s  =  5)  and  250  seconds 
(s  =  0.6)  averages  are  shown  on  Fig.  5.  The  first  is  far  too  severe  an 
assumption  for  operator  performance,  giving  NC's  nearly  double  those 
actually  observed  (and  those  given  by  theory  for  a  30-second  constant 
return  time).  The  250-second  average  return,  however,  lies  only  slightly 
above  the  30-second  constant  return  curve  and  is  in  good  agreement  with 
the  data.  Although  not  logically  an  adequate  formula  for  interpreting 
Peg  Count  and  Overflow  registrations  on  T-engineered  groups  under 
operator  dialing  conditions,  the  IF-formula  apparently  could  be  used  for 
this  purpose  with  suitable  s-values  determined  empirically. 

3.1.  Comparison  of  Some  Formulas  for  Estimating  Customers'  NC  Service 

on  Congested  Groups 

,     1 
As  has  been  previously  observed,  a  large  proportion  of  customers  who 

receive  a  busy  signal,  return  within  a  few  minutes  (on  Fig.  6,  75  per  cent 
of  the  customers  returned  within  10  minutes).  It  is  well  known  too,  that 
under  adverse  service  conditions  subscriber  attempts  (to  reach  a  par- 
ticular distant  office  for  example)  tend  to  produce  an  inflated  estimate 
of  the  true  offered  load.  A  count  of  calls  carried  (or  a  direct  measurement 
of  load  carried)  will  commonly  be  a  closer  estimate  of  the  offered  load 
than  a  count  of  attempts.  An  exception  may  occur  when  a  large  propor- 
tion of  attempts  is  lost,  indicating  an  offered  load  possibly  in  excess  even 
of  the  number  of  paths  provided.  Under  the  latter  condition  it  is  diffi- 
cult to  estimate  the  true  offered  load  by  any  method,  since  not  all  the 
attempts  can  be  expected  to  return  repeatedly  until  served;  instead,  a 
significant  number  will  be  abandoned  somewhere  through  the  trials.  In 
most  other  circumstances,  however,  the  carried  load  will  prove  a  reason- 
ably good  estimate  of  the  true  offered  load  in  systems  not  provided  with 
alternate  paths. 

This  is  a  matter  of  especial  interest  for  both  toll  and  local  operation 
in  America  since  principal  future  reliance  for  load  measurement  is  ex- 


THEORIES   FOR   TOLL   TRAFFIC   ENGINEERING   IN   THE   U.   S.    A.      435 

pected  to  be  placed  on  automatically  processed  TUR  data,  and  as  the 
TUR  is  a  switch  counting  device  the  results  will  be  in  terms  of  load 
carried.  Moreover,  the  quantity  now  obtained  in  many  local  exchanges 
is  load  carried.*  Visual  switch  counting  of  line  finders  and  selectors  off- 
normal  is  widely  practiced  in  step-by-step  and  panel  offices;  a  variety  of 
electromechanical  switch  counting  devices  is  also  to  be  found  in  crossbar 
offices.  It  is  common  to  take  load-carried  figures  as  equal  to  load-offered 
when  using  conventional  trunking  tables  to  ascertain  the  proper  pro- 
vision of  trunks  or  switches.  Fig.  7  compares  the  NC  predictions  made  by 
a  number  of  the  available  load-loss  formulas  when  load  carried  is  used  as 
the  entry  variable. 

The  lowest  curves  (1)  on  Fig.  7  are  from  the  Erlang  lost  calls  formula 
El  (or  B)  with  load  carried  L  used  as  the  offered  load  a.  At  low  losses, 
say  0.01  or  less,  either  L  or  a  =  L/[l  —  Ei(a)]  can  be  used  indiscrimi- 
nately as  the  entry  in  the  Ei  formula.  If  however  considerably  larger 
losses  are  encountered  and  calls  are  not  in  reality  "cleared"  upon  meet- 
ing NC,  it  will  no  longer  be  satisfactory  to  substitute  L  for  a.  In  this 
circumstance  it  is  common  to  calculate  a  fictitious  load  a'  to  submit  to 
the  c  paths  such  that  the  load  carried,  a'[I  —  Ei^dd')],  equals  the  desired 
L.  (This  was  the  process  used  in  Section  2  to  obtain  "  %  NC  existing.") 
The  curves  (2)  on  Fig.  7  show  this  relation ;  physically  it  corresponds  to 
an  initially  offered  load  of  L  erlangs  (or  L  call  arrivals  per  average  hold- 
ing time),  whose  overflow  calls  return  again  and  again  until  successful 
but  without  disturbing  the  randomness  of  the  input.  Thus  if  the  loss 
from  this  enhanced  random  traffic  is  E,  then  the  total  trials  seen  per 
holding  time  will  be  L(l  +  ^  +  ^'  -f  •  •  •)  =  L/(l  -  E)  =  a',  the  ap- 
parent arrival  rate  of  new  calls,  but  actually  of  new  calls  plus  return 
attempts. 

The  random  resubmission  of  calls  may  provide  a  reasonable  descrip- 
tion of  operation  under  certain  circumstances,  presumably  when  re-trials 
are  not  excessive.  Kosten^  has  discussed  the  dangers  here  and  provided 
upper  and  lowxr  limit  formulas  and  curves  for  estimating  the  proportions 
of  NC's  to  be  expected  when  re-trials  are  made  at  any  specified  fixed 
leturn  time.  His  lower  bounds  (lower  bound  because  the  change  in  con- 
gestion character  caused  by  the  returning  calls  is  ignored)  are  shown  by 
open  dots  on  Fig.  7  for  return  times  of  1.67  holding  times.  They  lie  above 
curves  (2)  (although  only  very  slightly  because  of  the  relatively  long 
return  time)  since  they  allo\\-  for  the  fact  that  a  call  shortly  returning 


*  In  fact,  it  is  difficult  to  see  how  any  estimate  of  offered  load,  other  than  carried 
load,  can  be  obtained  with  useful  reliability. 


436  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    1956 

after  meeting  a  busy  signal  will  have  a  higher  probability  of  again  find- 
ing all  paths  busy,  than  would  a  randomly  originated  call. 

The  curves  (3)  show  the  TF-formula  previously  developed  in  this  sec- 
tion, which  contemplates  exponential  return  times  on  all  NC  attempts. 
The  average  return  time  here  is  also  taken  as  1 .67  holding  times.  These 
curves  lie  higher  than  Kosten's  values  for  two  reasons.  First,  the  altered 
congestion  due  to  return  calls  is  allowed  for;  and  second,  with  exponential 
returns  nearly  two-thirds  of  the  return  times  are  shorter  than  the  aver- 
age, and  of  these,  the  shortest  ones  will  have  a  relatively  high  probability 
of  failure  upon  re-trying.  If  the  customers  were  to  return  with  exponen- 
tial times  after  waiting  an  average  of  only  0.2  holding  time  (e.g.,  30 
seconds  wait  for  150-second  calls)  the  TT^-curves  would  rise  markedly  to 
the  positions  shown  by  (4). 

Curves  (5)  and  (6)  give  the  proportions  of  time  that  all  paths  are  busy 
(equation  4)  under  the  T'F-formula  assumptions  corresponding  to  NC 
curves  (3)  and  (4)  respectively;  their  upward  displacement  from  the 
random  return  curves  (2)  reflects  the  disturbance  to  the  group  congestion 
produced  by  the  non-random  return  of  the  delayed  calls.  (The  limiting 
position  for  these  curves  is,  of  course,  given  by  Erlang's  E2  (or  C)  delay 
formula.)  As  would  be  expected,  curve  (6)  is  above  (5)  since  the  former 
contemplates  exponential  returns  with  average  of  0.2  holding  time,  as 
against  1.67  for  curve  (5).  Neither  the  (5)-curves  nor  the  open  dots  of 
constant  30-second  return  times  show  a  marked  increase  over  curves  (2). 
This  appears  to  explain  why  the  relationship  of  load  carried  versus  "NC 
existing"  (as  charted  in  Figs.  3  and  4)  was  found  so  insensitive  to  vari- 
able operating  procedures  in  handling  subsequent  attempts  in  toll  ring- 
down  operation,  and  again,  why  it  did  not  appreciably  change  under 
operator  dialing. 

Finally,  through  the  two  fields  of  curves  on  Fig.  7  is  indicated  the 
Poisson  summation  P{c,  L)  with  load  carried  L  used  as  the  entering 
variable.  The  fact  that  these  values  approach  closely  the  (2)  and  (3)  sets 
of  curves  over  a  considerable  range  of  NC's  should  reassure  those  who 
have  been  concerned  that  the  Poisson  engineering  tables  were  not  useful 
for  losses  larger  than  a  few  per  cent.* 

4.    SERVICE  REQUIREMENTS  FOR  DIRECT  DISTANCE  DIALING  BY  CUSTOMERS 

As  shown  by  the  TF-curves  (3)  on  Fig.  7,  the  attempt  failures  by  cus- 
tomers resulting  from  their  tendency  to  re-try  shortly  following  an  NC 

*  Reference  may  be  made  also  to  a  throwdown  by  C.  Clos  (Ref.  3)  using  the 
return  times  of  Fig.  6;  his  "%  NC"  results  agreed  closely  with  tlie  Poisson  pre- 
dictions. 


THEORIES   FOR   TOLL   TRAFFIC    ENGINEERING   IN   THE   U.    S.   A.      437 

would  be  expected  to  exceed  slightly  the  values  for  completely  random 
re-trials.  These  particular  curves  are  based  on  a  re-trial  interval  of  1.67 
times  the  average  circuit-use  time.  Such  moderation  on  the  part  of  the 
customer  is  probably  attainable  through  instructional  literature  and 
other  means  if  the  customer  believes  the  "NC"  or  "busy"  to  be  caused 
by  the  called  party's  actually  using  his  telephone  (the  usual  case  in  local 
practice).  It  would  be  considerably  more  difficult,  however,  to  dissuade 
the  customer  from  re-trying  at  a  more  rapid  rate  if  the  circuit  NC's 
should  generally  approach  or  exceed  actual  called-party  busies,  a  con- 
dition of  which  he  would  sooner  or  later  become  aware.  His  attempts 
might  then  be  more  nearly  described  by  the  (4)  curves  on  Fig.  7  cor- 
responding to  an  average  exponential  return  of  only  0.2  holding  time — or 
e\en  higher.  Such  a  result  would  not  only  displease  the  user,  but  also 
result  in  the  requirement  of  increased  switching  control  equipment  to 
handle  many  more  wasted  attempts. 

If  subscribers  are  to  be  given  satisfactory  direct  dialing  access  to  the 
iiitertoll  trunk  network,  it  appears  then  that  the  probability  of  finding 
XC  even  in  the  busy  hours  must  be  kept  to  a  low  figure.  The  following 
engineering  objective  has  tentatively  been  selected:  The  calls  offered  to 
the  ^'final"  group  of  trunks  in  an  alternate  route  system  should  receive  no 
more  than  3  per  cent  NC(P.03)  during  the  network  busy  season  busy  hour. 
(If  there  are  no  alternate  routes,  the  direct  group  is  the  "final"  route.) 

Since  in  the  nationwide  plan  there  will  be  a  final  route  between  each 
of  some  2,600  toll  centers  and  its  next  higher  center,  and  the  majority 
of  calls  offered  to  high  usage  trunks  will  be  carried  without  trying 
their  final  route  (or  routes),  the  over-all  point-to-point  service,  while 
not  easy  to  estimate,  will  apparently  be  quite  satisfactory  for  cus- 
tomer dialing. 

5.    ECONOMICS    OF   TOLL   ALTERNATE    ROUTING 

In  a  general  study  of  the  economics  of  a  nationwide  toll  switching  plan, 
made  some  years  ago  by  engineers  of  the  American  Telephone  and  Tele- 
graph Company,  it  was  concluded  that  a  toll  line  plant  sufficient  to  give 
ihe  then  average  level  of  service  (about  T-40)  with  ordinary  single-route 
procedures  could,  if  operated  on  a  multi-alternate  route  basis,  give  the 
desired  P.03  service  on  final  routes  with  little,  if  any,  increase  in  toll  line 
investment.*  On  the  other  hand  to  attain  a  similar  P.03  grade  of  service 
by  liberalizing  a  typical  intertoll  group  of  10  trunks  working  presently 


*  This,  of  course,  does  not  reflect  the  added  costs  of  the  No.  4  switching  equip- 
I  nient. 


438  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 

at  a  T-40  grade  of  service  and  an  occupancy  of  0.81  would  recjuire  an 
increase  of  43  per  cent  (to  14.3  trunks),  with  a  corresponding  decrease 
in  occupancy  to  0.57.  The  possible  savings  in  toll  lines  with  alternate 
routing  are  therefore  considerable  in  a  system  which  must  pro\'ide  a 
service  level  satisfactory  for  customer  dialing. 

In  order  to  take  fullest  advantage  of  the  economies  of  alternate  rout- 
ing, present  plans  call  for  five  classes  of  toll  offices.  There  will  be  a  large 
number  of  so-called  End  Offices,  a  smaller  number  of  Toll  Centers,  and 
progressively  fewer  Primary  Centers  (about  150),  Sectional  Centers 
(about  40)  and  Regional  Centers  (9),  one  of  which  will  be  the  National 
Center,  to  be  used  as  the  "home"  switching  point  of  the  other  eight 
Regional  Centers.*  Primary  and  higher  centers  will  be  arranged  to  per- 
form automatic  alternate  routing  and  are  called  Control  Switching 
Points  (CSP's).  Each  class  of  office  will  "home"  on  a  higher  class  of 
office  (not  necessarily  the  next  higher  one) ;  the  toll  paths  between  them 
are  called  "final  routes."  As  described  in  Section  4,  these  final  routes  will 
be  provided  to  give  low  delays,  so  that  between  each  principal  toll  point 
and  ever}'  other  one  there  will  be  available  a  succession  of  approximatelj' 
P.03  engineered  trunk  groups.  Thus  if  the  more  direct  and  heavily  loaded 
interconnecting  paths  commonly  provided  are  busj-  there  will  still  be  a 
good  chance  of  making  immediate  connection  over  final  routes. 

Fig.  8  illustrates  the  manner  in  which  automatic  alternate  routing  will 
operate  in  comparison  with  present-day  operator  routing.  On  a  call  from 
Syracuse,  X.  Y.,  to  Miami,  Florida,  (a  distance  of  some  1,250  miles), 
under  present-day  operation,  the  Syracuse  operator  signals  Albany,  and 
requests  a  trunk  to  Miami.  With  T-schedule  operation  the  Syracuse- 
Miami  traffic  might  be  expected  to  encounter  as  much  as  25  per  cent  NC 
during  the  busy  hour,  and  approximately  4  per  cent  NC  for  the  whole 
day,  producing  perhaps  a  two-minute  over-all  speed  of  serA-ice  in  the 
busy  season. 

With  the  proposed  automatic  alternate  routing  plan,  all  points  on  the 
chart  will  have  automatic  switching  systems. f  The  customer  (or  the 
operator  until  customer  dialing  arrangements  are  completed)  will  dial  a 
ten-digit  code  (three-digit  area  code  305  for  Florida  plus  the  listed 
Miami  seven-digit  telephone  number)  into  the  Jiiachine  at  Syracuse. 
The  various  routes  which  then  might  conceivably  be  tried  automatically 


*  Sec  the  hihlio^rajjliy  ( i);irticulMily  Pilliod  and  Truitt)  for  details  of  tlie 
general  trunkinji  plan. 

t  The  notation  uscmI  on  the  diagram  of  Fig.  8  is:  Opon  firclo  —  Primary  Center 
(Syracuse,  Miamij;  Triangle  —  Sectional  Center  (All)an\-,  Jacksonville);  Sqviare 
—  Regional  Center  (White  Plains,  Atlanta,  St.  Louis;  St.  Louis  is  also  the  Na- 
tional Center). 


THEORIES   FOR   TOLL   TRAFFIC    ENGINEERING   IN   THE   U.    S.    A.      439 


PRESENT  OPERATOR 

ROUTING  '^^ 


AUTOMATIC  ALTERNATE 
ROUTING 


white   Plains 
N.  Y.) 


Miami 


Miami 


Fig.  8  —  Present  and  proposed  methods  of  handling  a  call  from  Syracuse,  N.  Y., 
to  Miami,  Florida. 


are  shown  on  the  diagram  numbered  in  the  order  of  trial;  in  this  par- 
ticular layout  shown,  a  maximum  of  eleven  circuit  groups  could  be  tested 
for  an  idle  path  if  each  high  usage  group  should  be  found  NC.  Dotted 
lines  show  the  high  usage  roiites,  which  if  found  busy  will  overflow  to  the 
final  groups  represented  by  solid  lines.  The  switching  ecjuipment  at  each 
point  upon  finding  an  idle  circuit  passes  on  the  required  digits  to  the 
next  machine. 

While  the  routing  possibilities  shown  are  factual,  only  in  rare  instances 
would  a  call  be  completed  over  the  final  route  via  St.  Louis.  Even  in  the 
busy  season  busy  hour  just  a  small  portion  of  the  calls  would  be  expected 
to  be  switched  as  many  as  three  times.  And  only  a  fraction  of  one  per 
cent  of  all  calls  in  the  busy  hour  should  encounter  NC.  As  a  result  the 
service  will  be  fast.  When  calls  are  handled  by  a  toll  operator,  the  cus- 


440 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL;    MARCH    1956 


tomer  will  not  ordinarily  need  to  hang  up  when  NC  is  obtained.  When 
he  himself  dials,  a  second  trial  after  a  short  wait  following  NC  should 
have  a  high  probability  of  success. 

Not  many  situations  will  be  as  complex  as  shown  in  Fig.  8;  commonly 
several  of  the  links  between  centers  will  be  missing,  the  particular  ones 
retained  having  been  chosen  from  suitable  economic  studies.  A  large 
number  of  switching  arrangements  Avill  be  no  more  involved  than  the 
illustrative  one  shown  in  Fig.  9(a),  centering  on  the  Toll  Center  of 
Bloomsburg,  Pennsylvania.  The  dashed  lines  indicate  high  usage  groups 
from  Bloomsburg  to  surrounding  toll  centers;  since  Bloomsburg  "homes" 
on  Scranton  this  is  a  final  route  as  denoted  by  the  solid  line.  As  an  exam- 
ple of  the  operation,  consider  a  call  at  Bloomsburg  destined  for  Williams- 
port.  Upon  finding  all  direct  trunks  busy,  a  second  trial  is  made  via 
Harrisburg;  and  should  no  paths  in  the  Harrisburg  group  be  available, 
a  third  and  final  trial  is  made  through  the  Scranton  group. 

In  considering  the  traffic  flow  of  a  network  such  as  illustrated  at 
Bloomsburg  it  is  convenient  to  employ  the  conventional  form  of  a  two- 
stage  graded  multiple  having  "legs"  of  varying  sizes  and  traffic  loads 
individual  to  each,  as  shown  in  Fig.  9(b).  Here  only  the  circuits  im- 
mediately outgoing  from  the  toll  center  are  shown;  the  parcels  of  traffic 


(a)  GEOGRAPHICAL    LAYOUT 
WILLIAMSPORT  I 


SCRANTON 


BLOOMSBURG 
HARRISBURG  PA. 

(b)  GRADED    MULTIPLE   SCHEMATIC 


FRACKVILLE 

HAZLETON 

WILKES- 
BARRE 


PHILADELPHIA 


FINAL    GROUP    TO   SCRANTON 


H.U.  GROUP   TO   HARRISBURG 


.1    M    t 


I 


NO.  TRUNKS   IN   H.U.  GROUPS  I      [T]  [jF]  [^  [A]  [T]  [28 1  rsl  m 

LOAD    TO    AND  FROM  ^^^      .^.      ^^     ^  

DISTANT   OFFICE    (CCS)  "^^^  '^'     ^^    ^'^^    ^^'     '^0    '^3   836   228    154 

DISTANT   OFFICE  SCRN  HBG  PTVL   SHKN  SNBY  WMPT  FKVL    HZN  WKSB  PHLA 


Fig.  9 
liiirg,  Pa. 


Aulonialic  ;ilU'riiaie  routing  for  direct  distance  dialing  at  Blooms- 


THEORIES   FOR   TOLL   TRAFFIC   ENGINEERING   IN    THE   U.    S.   A.      441 

calculated  for  each  further  connecting  route  will  be  recorded  as  part  of 
the  offered  load  for  consideration  when  the  next  higher  switching  center 
is  engineered.  It  is  implicitly  assumed  that  a  call  which  has  selected  one 
of  the  alternate  route  paths  will  be  successful  in  finding  the  necessary 
paths  available  from  the  distant  switching  point  onward.  This  is  not 
quite  true  but  is  believed  generally  to  be  close  enough  for  engineering 
piu'poses,  and  permits  ignoring  the  return  attempt  problem. 

6.    NEW  PROBLEMS  IN  THE  ENGINEERING  AND  ADMINISTRATION  OF  INTER- 
TOLL   GROUPS   RESULTING   FROM   ALTERNATE   ROUTING 

With  the  greatly  increased  teamwork  among  groups  of  intertoll  trunks 
which  supply  overflow  calls  to  an  alternate  route,  an  unexpected  increase 
or  flurry  in  the  offered  load  to  any  one  can  adversely  affect  the  service  to 
all.  The  high  efficiency  of  the  alternate  route  networks  also  reduces  their 
overload  carrying  ability.  Conversely,  the  influence  of  an  underprovision 
of  paths  in  the  final  alternate  route  may  be  felt  by  many  groups  which 
overflow  to  it.  With  non-alternate  route  arrangements  only  the  single 
groups  having  these  flurries  would  be  affected. 

Administratively,  an  alternate  route  trunk  layout  may  well  prove 
easier  to  monitor  day  by  day  than  a  large  number  of  separate  and  in- 
dependent intertoll  groups,  since  a  close  check  on  the  service  given  on 
the  final  routes  only  may  be  sufficient  to  insure  that  all  customers  are 
being  served  satisfactorily.  When  rearrangements  are  indicated,  how- 

SIMPLE  PROGRESSIVE 

GRADED  MULTIPLE  GRADED  MULTIPLE 

(a)  (b) 


t      t      t     t  t  t  tt    t  t  tl 

ILLUSTRATIVE    INTERLOCAL    AND  INTERTOLL 
ALTERNATE    ROUTE    TRUNKING    ARRANGEMENT; 

(c)  (d) 


t    t      t   t    t  =    ,-""    ^ 

tttl It  ttl   1   t 

Fig.  10  —  Graded  multi])los  .•nid  altornaic  route  trunking  nrrangeinoiits. 


I 


442  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 

ever,  the  determination  of  the  proper  place  to  take  action,  and  the  de 
sirable  extent,  may  sometimes  be  difficult  to  determine.  Suitable  traffic 
measuring  devices  must  be  provided  with  these  latter  problems  in  mind 
For  engineering  purposes,  it  will  be  highly  desirable: 

(1)  To  be  able  to  estimate  the  load-service  relationships  with  any 
specified  loads  offered  to  a  particular  intertoll  alternate  routing  network; 
and 

(2)  To  know  the  day-to-day  busy  hour  variations  in  the  various 
groups'  offered  loads  during  the  busy  season,  so  that  the  general  grade  of 
service  given  to  customers  can  be  estimated. 

The  balance  of  this  paper  will  review  the  studies  which  have  been  made 
in  the  Bell  System  toward  a  practicable  method  for  predicting  the  grade 
of  service  given  in  an  alternate  route  network  under  any  given  loads. 
Analyses  of  the  day-to-day  load  variations  and  their  effects  on  customer 
dialing  service  are  currently  being  made,  and  will  be  reported  upon  later. 

?; 

7.    LOAD-SERVICE   RELATIONSHIPS  IN  ALTERNATE    ROUTE   SYSTEMS 

In  their  simplest  form,  alternate  route  systems  appear  as  symmetrical 
graded  multiples,  as  shown  in  Fig.  10(a)  and  10(b).  Patterns  such  as 
these  have  long  been  used  in  local  automatic  systems  to  partially  over- 
come the  trunking  efficiency  limitations  imposed  by  limited  access 
switches.  The  traffic  capacity  of  these  arrangements  has  been  the  sub- 
ject of  much  study  by  theory  and  "throwdowns"  (simulated  traffic 
studies)  both  in  the  United  States  and  abroad.  Field  trials  have  sub- 
stantiated the  essential  accuracy  of  the  trunking  tables  which  have 
resulted. 

In  toll  alternate  route  systems  as  contemplated  in  America,  however, 
there  will  seldom  be  the  symmetry  of  pattern  found  in  local  graded 
multiples,  nor  does  maximum  switch  size  generally  produce  serious 
limitation  on  the  access.  The  ''legs"  or  first-choice  trunk  groups  will  vary 
widely  in  size;  likewise  the  number  of  such  groups  overflowing  calls 
jointly  to  an  alternate  route  may  cover  a  considerable  range.  In  all  cases 
a  given  group,  whether  or  not  a  link  of  an  alternate  route,  will  have  one 
or  more  parcels  of  traffic  for  which  it  is  the  first-choice  route.  [See  the 
right-hand  parcel  of  offered  traffic  on  Fig.  10(c).]  Often  this  first  routed 
traffic  will  Ijc  the  bulk  of  the  load  offered  to  the  group,  which  also  serves 
as  an  alternate  I'oute  for  other  traffic. 

The  simplest  of  the  approximate  formulas  developed  for  solving  the 
local  graded  multiple  problems  are  hopelessly  unwieldy  when  applied 
to  such  arrangements  as  shown  in  Fig.  10(d).  Likewise  it  is  impracticable  i 


THEORIES   FOR  TOLL   TRAFFIC    ENGINEERING   IN   THE   U.    S.    A.      443 

to  solve  more  than  a  few  of  the  infinite  variety  of  arrangements  by  means 

of  "throwdowns." 

However,  for  both  engineering  (planning  for  future  trunk  provisions) 
I  and  administration  (current  operating)  of  trunks  in  these  multi-alternate 

routing  systems,  a  rapid,  simple,  but  reasonably  accurate  method  is 
(required.  The  basis  for  the  method  which  has  been  evolved  for  Bell 

System  use  will  be  described  in  the  following  pages. 

7.1.  The  "Peaked"  Character  of  Overflow  Traffic 

The  difficulty  in  predicting  the  load-service  relationship  in  alternate 
route  systems  has  lain  in  the  non-random  character  of  the  traffic  over- 
flowing a  first  set  of  paths  to  which  calls  may  have  been  randomly 
offered.  This  non-randomness  is  a  well  appreciated  phenomenon  among 
traffic  engineers.  If  adecjuate  trunks  are  provided  for  accommodating 
the  momentary  traffic  peaks,  the  time-call  level  diagram  may  appear 
as  in  Fig.  11(a),  (average  level  of  9.5  erlangs).  If  however  a  more  limited 
j  number  of  trunks,  say  a:  =  12,  is  provided,  the  peaks  of  Fig.  11(a)  will  be 
Ichpped,  and  the  overflow  calls  will  either  be  "lost"  or  they  may  be 
j  handled  on  a  subsequent  set  of  paths  y.  The  momentary  loads  seen  on  2/ 
then  appear  as  in  Fig.  11(b).  It  will  readily  be  seen  that  a  given  average 
i  load  on  the  y  trunks  will  have  quite  different  fluctuation  characteristics 
i  than  if  it  had  been  found  on  the  x  trunks.  There  will  be  more  occurrences 
of  large  numbers  of  calls,  and  also  longer  intervals  when  few  or  no  calls 
are  present.  This  gives  rise  to  the  expression  that  overflow  traffic  is 
"peaked." 

Peaked  traffic  requires  more  paths  than  does  random  traffic  to  operate 
at  a  specified  grade  of  delayed  or  lost  calls  service.  And  the  increase  in 
paths  required  will  depend  upon  the  degree  of  peakedness  of  the  traffic 
involved.  A  measure  of  peakedness  of  overflow  traffic  is  then  required 
which  can  be  easily  determined  from  a  knowledge  of  the  load  offered  and 
the  number  of  trunks  in  the  group  immediately  available. 

In  1923,  G.  W.  Kendrick,  then  with  the  American  Telephone  and 
I  Telegraph  Company,  undertook  to  solve  the  graded  multiple  problem 
■through  an  application  of  Erlang's  statistical  eciuilibrium  method.  His 
i  principal  contribution  (in  an  unpublished  memorandum)  was  to  set  up 
I  the  equations  for  describing  the  existence  of  calls  on  a  full  access  group 
\oi  X  -{-  y  paths,  arranged  so  that  arriving  calls  always  seek  service  first 
iu  the  .T-group,  and  then  in  the  ^/-group  when  the  x  are  all  busy. 

Let  f{m,  n)  be  the  probability  that  at  a  random  instant  m  calls  exist 
j  on  the  x  paths  and  n  calls  on  the  y  paths,  when  an  average  Poisson  load 


444 


THE   BELL   SYSTEM   TECHNICAL  JOURNAL,   MARCH    1956 


of  a  erlangs  is  submitted  to  the  x  -\-  y  paths.  The  general  state  equation 
for  all  possible  call  arrangements,  is 


(a*  +  m  +  n)f{m,  n)  =  (w  +  l)/(m  +  1,  n) 

+  (n  +  l)/(m,  w  +  1)  +  ajim  —  1,  n)  +  aj{x,  n  —  1)% 


(7) 


in  which  the  term  marked  {%)  is  to  be  included  only  when  m  =  x,  and 
*  indicates  that  the  a  in  this  term  is  to  be  omitted  when  in  -\-  n  =  x  -{-  y. 
m  and  n  may  take  values  only  in  the  intervals,  0  -^  m  ^  x;Q  -^  n  -^  y. 
As  written,  the  equation  represents  the  "lost  calls  cleared"  situation. 


(a)  RANDOM    TRAFFIC 


10  00  AM 


<  I 

if)  Q. 


a. 


2 
to 


10  00  A  M 


10  30 
TIME    OF    DAY 


(b)    PEAKED  OVERFLOW    TRAFFIC 


PI 

-^ 

10  30 
TIME   OF    DAY 


Fig.  11  —  Production  of  peakedness  in  overflow  traffic. 


THEORIES   FOR   TOLL   TRAFFIC   ENGINEERING   IN   THE   U.    S.   A.      445 

By  choosing  x  -]-  y  large  compared  with  the  submitted  load  a  a  "lost 
calls  held"  situation  or  infinite-overflow-trunks  result  can  be  approached 
as  closely  as  desired. 

Kendrick  suggested  solving  the  series  of  simultaneous  equations  (7)  by 
determinants,  and  also  by  a  method  of  continued  fractions.  However 
little  of  this  numerical  work  was  actually  undertaken  until  several  years 
later. 

Early  in  1935  Miss  E.  V.  Wyckoff  of  Bell  Telephone  Laboratories  be- 
came interested  in  the  solution  of  the  (x  -\-  1)(^/  +  1)  lost  calls  cleared 
simultaneous  equations  leading  to  all  terms  in  the  /(m,  n)  distribution. 
She  devised  an  order  of  substituting  one  equation  in  the  next  which  pro- 
vided an  entirely  practical  and  relatively  rapid  means  for  the  numerical 
solution  of  almost  any  set  of  these  equations.  By  this  method  a  con- 
siderable number  of  /(m,  n)  distributions  on  x,  y  type  multiples  with 
varying  load  levels  were  calculated. 

From  the  complete  m,  n  matrix  of  probabilities,  one  easily  obtains  the 
distribution  9m{n)  of  overflow  calls  when  exactly  m  are  present  on  the 
lower  group  of  x  trunks;  or  by  summing  on  m,  the  d{n)  distribution  with- 
out regard  to  m,  is  realized.  A  number  of  other  procedures  for  obtaining 
the/(m,  n)  values  have  been  proposed.  All  involve  lengthy  computations, 
very  tedious  for  solution  by  desk  calculating  machines,  and  most  do  not 
have  the  ready  checks  of  the  WyckofT-method  available  at  regular  points 
through  the  calculations. 

In  1937  Kosten^  gave  the  following  expression  for  /(m,  n) : 

/(»,  n)  =  (-  l)V.fe)  i  (i)  M^-      "f^'l.,  (8) 


i=0 


(Pi^l{x)ipi(x) 


where 


(po{x)  = 


x^—a 

a  e 


xl 


;  and  for  i  >  0, 


;=o  \         J         /  (.^•  -  J)i 


These  equations,  too,  are  laborious  to  calculate  if  the  load  and  num- 
1  K^rs  of  trunks  are  not  small.  It  would,  of  course,  be  possible  to  program  a 
modern  automatic  computer  to  do  this  work  with  considerable  rapidity. 

The  corresponding  application  of  the  statistical  equilibrium  equations 
to  the  graded  multiple  problem  was  visualized  by  Kendrick  who,  how- 
ever, went  only  so  far  as  to  write  out  the  equation  for  the  three-trunk 


446  THE   BELL   SYSTEM   TECHNICAL  JOURNAL,    MARCH    1956 

case  consisting  of  two  subgroups  of  one  trunk  each  and  one  common 
overflow  trunk. 

Instead  of  solving  the  enormously  elaborate  system  of  equations  de- 
scribing all  the  calls  which  could  simultaneously  be  present  in  a  large 
multiple,  several  ingenious  methods  of  convoluting  the 

X 

6(n)  =   Z/(w,  n) 

overflow  distributions  from  the  individual  legs  of  a  graded  multiple  have 
been  devised.  For  example,  for  the  multiple  of  Fig.  10(a),  the  probability 
of  loss  Pi  as  seen  by  a  call  entering  subgroup  number  i,  is  approximately, 

Pi  =  2  £  e.Ar)-rl^{z  -r)  +J:  d.Ar)  (9) 

r=0  z=y  T—y 

in  which  \l/{z  —  r)  is  the  probability  of  exactly  z  —  r  overflow  calls  being 
present,  or  wanting  to  be  present,  on  the  alternate  route  from  all  the 
subgroups  except  the  zth,  and  with  no  regard  for  the  numbers  of  calls 
present  in  these  subgroups.  The  ^x,i(^)  =  jiixi ,  r)  term,  of  course,  con- 
templates all  paths  in  the  particular  originating  call's  subgroup  being 
occupied,  forcing  the  new  call  arriving  in  subgroup  i  to  advance  to  the 
alternate  route.  This  corresponds  to  the  method  of  solving  graded  mul- 
tiples developed  by  E.  C.  Molina^  but  has  the  advantage  of  overcoming 
the  artificial  "no  holes  in  the  multiple"  assumption  which  he  made. 
Similar  calculating  procedures  have  been  suggested  by  Kosten.*  These 
computational  methods  doubtless  yield  useful  estimates  of  the  resulting 
service,  and  for  the  limited  numbers  of  multiple  arrangements  which 
might  occur  in  within-office  switching  trains  (particularly  ones  of  a  sym- 
metrical variety)  such  procedures  might  be  practicable.  But  it  would  be 
far  too  laborious  to  obtain  the  individual  overflow  distributions  Q{n), 
and  then  convolute  them  for  the  large  variety  of  loads  and  multiple 
arrangements  expected  to  be  met  in  toll  alternate  routing. 

7.2.  Approximate  Description  of  the  Character  of  Overflow  Traffic 

It  was  natural  that  various  approximate  procedures  should  be  tried  in 
the  attempt  to  obtain  solutions  to  the  general  loss  formula  sufficiently 
accurate  for  engineering  and  study  purposes.  The  most  ol^vious  of  these 
is  to  calculate  the  lower  moments  or  semi-invariants  of  the  loads  over- 
flowing th(;  sul)groups,  and  from  them  construct  approximate  fitting 

*  Kosten  gives  the  above  approximation  (9),  which  he  calls  Wb^,  Jis  an  upper 
limit  to  the  blocking.  He  also  gives  a  lower  limit ,  Wr,  in  which  z  =  //  throughout 
(References  4,  5). 


THEORIES   FOR   TOLL   TRAFFIC   ENGINEERING   IN   THE   U.    S.    A.      447 

I  distributions  for  6{n)  mid  dx(;n).  Since  each  such  overflow  is  independent 
I  of  the  others,  they  may  be  combined  additively  (or  convokited),  to  ob- 
[tain  the  corresponchng  total  distribution  of  calls  appearing  before  the 
,  I  alternate  route  (or  common  group) .  It  may  further  be  possible  to  obtain 
I  [an  approximate  fitting  distribution  to  the  sum-distribution  of  the  over- 
flow calls. 

The  ordinary  moments  about  the  0  point  of  the  subgroup  overflow 
distribution,  when  m  of  the  x  paths  are  busy,  are  found  by 

V 

ta'im)  =   2  njim,  n)  (10) 

When  an  infinite  number  of  |/-paths  is  assumed,  the  resulting  expres- 
sions for  the  mean  and  variance  are  found  to  be:* 
Number  of  x-paths  busy  unspecified :'\ 

Mean  =  a  =  a-Ei,^{a)  (11) 

Variance  =  v  =  a[l  —  a  -{-  a{x  -\-  I  -\-  a  —  a)'^]  (12) 

All  x-paths  occupiedi 

Mean  =  a^^  =  a[x  -  a  +  1  -\-  aEiMf^  (13) 

Variance  =  v^  =  ax[l  —  ax  +  2a(x  +  2  +  a^  —  a)~^]  (14) 

Equations  (11)  and  (12)  have  been  calculated  for  considerable  ranges 
1  of  offered  load  a  and  paths  x.  Figs.  12  and  13  are  graphs  of  these  results. 
i  For  example  when  a  load  of  4  erlangs  is  submitted  to  5  paths,  the  aver- 
I  age  overflow  load  is  seen  to  be  a  =  0.80  erlang,  the  same  value,  of 
I  course,  as  determined  through  a  direct  application  of  the  Erlang  Ei 
formula.  During  the  time  that  all  x  paths  are  busy,  however,  the  over- 
flow load  wdll  tend  to  exceed  this  general  level  as  indicated  by  the  value 
of  ax  =  1.41  erlangs  calculated  from  (13).  Similarly  the  variance  of  the 
overflow  load  will  tend  to  increase  when  the  x-paths  are  fully  occupied, 

*  The  derivation  of  these  equations  is  given  in  Appendix  I. 
t  The  skewness  factor  may  also  be  of  interest : 


ilz 


l^i: 


3/2 


^"  +  "-"^"'  +a^      (15) 


x+1  +a-  a  \x  +  2\{x-a)'^^-2{x-a)  +  x  +  2  +  {x^-2-a)a 

+  3(1  -a)   I  +  a(l  -  a)(l  -  2a) 


o 


K:i'  \ 


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448 


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452  THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 

as  shown  by  ?;  =  1.30,  and  Vx  =  1.95.  In  all  cases  the  variances  v  and  Vx 
will  exceed  the  variance  of  corresponding  Poisson  traffic  (which  would 
have  variances  of  a  and  ax  respectively). 

7.2.1.  A  Prohahility  Distribution  for  Overflow  Traffic 

It  would  be  of  interest  to  be  able,  given  the  first  several  descriptive 
parameters  of  any  traffic  load  (such  as  the  mean  and  variance  and  skew- 
ness  factors  of  the  overflow  from  a  group  of  trunks),  to  construct  an 
approximate  probability  distribution  d{n)  which  would  closely  describe 
the  true  momentary  distribution  of  simultaneous  calls.  Any  proposed 
fitting  distribution  for  the  overflow  from  random  traffic  offered  to  x 
trunks,  can,  of  course,  be  compared  with  .     ^ 

X 

determined  from  (7)  or  (8). 

Suitable  fitting  curves  should  give  probabilities  for  all  possitive  in- 
tegral values  of  the  variable  (including  zero) ,  and  have  sufficient  unspeci- 
fied constants  to  accommodate  the  parameters  selected  for  describing 
the  distribution.  Moreover,  the  higher  moments  of  a  fitting  distribution 
should  not  diverge  too  radically  from  those  of  the  true  distribution ;  that 
is,  the  "natural  shapes"  of  fitting  and  true  distributions  should  be  simi- 
lar. Particularly  desirable  would  be  a  fitting  distribution  form  derived 
with  some  attention  to  the  physical  circumstances  causing  the  ebb  and 
flow  of  calls  in  an  overflow  situation.  The  following  argument  and  der- 
ivation undertake  to  achieve  these  desiderata.* 

A  Poisson  distribution  of  offered  traffic  is  produced  by  a  random  arrival 
of  calls.  The  assumption  is  made  or  implied  that  the  probability  of  a  new 
arrival  in  the  next  instant  of  time  is  quite  independent  of  the  number 
currently  present  in  the  system.  When  this  randomness  (and  correspond- 
ing independence)  are  disturbed  the  resulting  distribution  will  no  longer 
be  Poisson.  The  first  important  deviation  from  the  Poisson  would  be 
expected  to  appear  in  a  change  from  variance  =  mean,  to  variance  ^ 


*  A  two-parameter  function  which  has  the  ability  to  fit  quite  well  a  wide  variety 
of  true  overflow  distributions,  has  the  form 

t(n)  =  Kin  +  l)''e-^(''+i) 

in  which  K  is  the  normalizing  constant.  The  distribution  is  displaced  one  unit 
from  the  usual  discrete  generalized  exponential  form,  so  that  ^(0)  9^  0.  The  ex- 
pression, however,  has  little  rationale  for  being  selected  a  priori  as  a  suitable 
fitting  function. 


I 


THEORIES    FOR   TOLL   TRAFFIC   ENGINEERING   IN   THE   U.    S.   A.      453 

mean.  Corresponding  changes  in  the  higher  moments  would  also  be 
expected. 

WTiat  would  be  the  physical  description  of  a  cause  system  with  a  vari- 
ance smaller  or  larger  than  the  Poisson?  If  the  variance  is  smaller,  there 
must  be  forces  at  work  which  retard  the  call  arrival  rate  as  the  number 
of  calls  recently  offered  exceeds  a  normal,  or  average,  figure,  and  which 
increase  the  arrival  rate  when  the  number  recently  arrived  falls  below 
the  normal  level.  Conversely,  the  variance  will  exceed  the  Poisson's 
.should  the  tendencies  of  the  forces  be  reversed.*  This  last  is,  in  fact,  a 
rough  description  of  the  incidence  rates  for  calls  overflowing  a  group  of 
trunks. 

Since  holding  times  are  attached  to  and  extend  from  the  call  arrival 
instants,  calls  are  enabled  to  project  their  influence  into  the  future;  that 
is,  the  presence  of  a  considerable  number  of  calls  in  a  system  at  any  in- 
stant reflects  their  having  arrived  in  recent  earlier  time,  and  now  can  be 
used  to  modify  the  current  rate  of  call  arrival. 

Let  the  probability  of  a  call  originating  in  a  short  interval  of  time  dt  be 

Po.n  =  [a  +  (n  —  a)co(n)]  dt 

where       n  =  number  of  calls  present  in  the  system  at  time  t, 

a  =  base  or  average  arrival  rate  of  calls  per  unit  time,  and 
w(n)  =  an  arbitrary  function  which  regulates  the  modification  in 
call  origination  rate  as  the  number  of  calls  rises  above 
or  falls  below  a. 
Correspondingly,  let  the  probability  that  one  of  n  calls  will  end  in  the 
short  interval  of  time  dt  be 

which  will  be  satisfied  in  the  case  of  exponential  call  holding  times,  with 
mean  unity.  Following  the  usual  Erlang  procedure,  the  general  statistical 
equilibrium  equation  is 


(16) 


Jin)    =   /(n)[(l     -    Po.n){l    -    Pe,n)\    +  /(«    "     l)Po,n-l(l    "    Pe.n-l) 

-Vj{n+  1)(1  -  Po.„+i)P,,„+i 
which  gives 

(Po,„  +  P.,„)/(n)  =  Po,«-i/(n  -  1)  +  Pe,n+xKn  +  1) 

i  ignoring  terms  of  order  higher  than  the  first  in  dt. 

*  The  same  thinking  lias  been  used  by  Vaiilot^  for  decreasing  the  call  arrival 
I  rate  according  to  the  number  momentaril}^  present;  and  by  Lundquist^  for  both 
increasing  and  decreasing  the  arrival  rate. 


454  THE   BELL   SYSTEM   TECHNICAL  JOURNAL,   MARCH    1956 


(17) 


Or, 

[a  +  (n  —  a)w(?i)  +  ??.]/(n) 

=  [a  +  (n  -a-  l)co(n  -  l)]/(/^  -  1)  +  (n  +  l)/(7i  +  1) 

The  choice  of  aj(n)  will  determine  the  solution  of  (17).  Most  simply, 
co(n)  =  k,  making  the  variation  from  the  average  call  arrival  rate  directly- 
proportional  to  the  deviation  in  numbers  of  calls  present  from  their 
average  number.  In  this  case,  the  solution  for  an  unlimited  trunk  group 
becomes,  with  a'  =  a{l  —  k), 

a  (a   +  k)  -■■  [a   +  {n  -  1)A;] 

fin)  =  


n! 


^^    ,    ,    a' (a'  +  k)    ,    a' (a'  +  k)(a'  +  2k)    , 
1  +  «   H ^t; H ^ TT, + 


(18) 


2!  '  3! 

which  may  also  be  written  after  setting  a"  =  a'/k  =  a(l  —  k)/k,  as 

a'ia'  +  1)  •  •  •  [a"  +  (n  -  1)]A;" 
fin)  =  


n! 


(19); 


(1  -  k)- 
The  generating  function  (g.f.)  of  (19)  is 


Z/(n)r  = 


(1  -  kT)-"" 


n=0  (1    -    k)--" 

which  is  recognized  as  that  for  the  negative  binomial,  as  distinguished 
from  the  g.f., 

P 


(i  +  ?  tX 


(1/g)^ 

for  the  positive  binomial. 

The  first  four  descriptive  parameters  of /(w)  are: 


Order 

Moment  about  Mean 

Descriptive  Parameter 

1 

Ml 

=  0 

Mean  =  n  =  a 

(20) 

2 

M2 

=  variance,  v  =  a/(l  —  k) 

Std  Devn,  <r  =  [a/(l  -  fc)]'/2 

(21) 

3 

f^a 

a(l  +  k) 
(1  -  fc)^ 

Skewness,  \/sT  —       —      ,  , 

(22) 

4 

M4 

3a2(l  -  A0  +  a(fc2+4A;  +  l) 

M4               A;2  +  4fc  +  1 

Kurtosis,  /3.,  =  -  =  3  +  —7^ ry- 

o-*                 a(l  —  k) 

(23) 

(1  -  fc)3 

I 


THEORIES   FOR   TOLL   TRAFFIC    ENGINEERING   IN   THE    U.    S.   A.      455 

Since  only  two  constants,  a  and  k,  need  specification  in  (18)  or  (19), 
the  mean  and  variance  are  sufficient  to  fix  the  distribution.  That  is,  with 
the  mean  /7  and  variance  v  known, 

a  =  ,7         or        a'  =  n(l  -  k)  =  if/v,        or        a"  =  n(l  -  k)/k     (24) 

A:  =  1  -  a/y  =  1  -  n/v.  (25) 

The  probability  density  distribution  f(n)  is  readily  calculated  from 
(19);  the  cumulative  distribution  G(^n)  also  may  be  found  through  use 
of  the  Incomplete  Beta  Function  tables  since 

G(^n)  =  hi7i  -  l,a") 

(26) 
=  h(n  -  l,a(l  -  k)/k) 

The  goodness  with  which  the  negative  binomial  of  (19)  fits  actual  dis- 
tributions of  overflow  calls  requires  some  investigation.  Perhaps  a  more 
elaborate  expression  for  co(n)  than  a  constant  k  in  (17)  is  required.  Three 
comparisons  appear  possible:  (1),  comparison  with  a  variety  of  0«(n) 
distributions  with  exactly  m  calls  on  the  x  trunks,  or  d{n)  with  m  unspeci- 
fied, (obtained  by  solving  the  statistical  equilibrium  equations  (7)  for  a 
divided  group) ;  (2),  comparison  with  simulation  or  "throwdown"  results; 
and  (3),  comparison  with  call  distributions  seen  on  actual  trunk  groups. 
These  are  most  easily  performed  in  the  order  listed.* 

Co7nparison  of  Negative  Binomial  with  True  Overflow  Distributions 

Figs.  14  to  17  show  various  comparisons  of  the  negative  binomial  dis- 
tribution with  true  overflow  distributions.  Fig.  14  gives  in  cumulative 
form  the  cases  of  5  erlangs  offered  to  1,  2,  5,  and  10  trunks.  The  true 


j  =  n 


distributions  (shown  as  solid  lines)  are  obtained  by  solving  the  difference 
equations  (7)  in  the  manner  described  in  Section  7.1.  The  negative  bi- 
nomial distributions  (shown  dashed)  are  chosen  to  have  the  same  mean 
and  variance  as  the  several  F{^n)  cases  fitted.  The  dots  shown  on 


*  Comparison  could  also  be  made  after  equating  means  and  variances  respec- 
tively, between  the  higher  moments  of  the  overflow  traffic  beyond  x  trunks  and 
the  corresponding  negative  binomial  moments:  e.g.,  the  skewness  given  by  (15) 
can  be  compared  with  the  negative  binomial  skewness  of  (22).  The  difficulty  here 
is  that  one  is  unable  to  judge  whether  the  disparity  between  the  two  distribution 
functions  as  described  by  differences  in  their  higher  parameters  is  significant  or 
not  for  traffic  engineering  purposes. 


456 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


the  figure  are  for  random  (Poisson)  traffic  having  the  same  mean  values 
as  the  /''  distributions.  The  negative  binomial  provides  excellent  fits 
down  to  cumulated  probabilities  of  0.01,  with  a  tendency  thereafter  to 
give  somewhat  larger  values  than  the  true  ones.  The  Poisson  agreement 
is  good  only  for  the  overflow  from  a  single  trunk,  as  might  have  been 
anticipated,  the  divergence  rapidly  increasing  thereafter. 

Fig.  15  corresponds  with  the  cases  of  Fig.  14  except  that  the  true  over- 
flow Fxi^n)  distributions  for  the  conditional  situation  of  all  .r-paths 
busy,  are  fitted.  Again  the  negative  binomial  is  seen  to  give  a  good  agree- 
ment down  to  0.01  probability,  with  somewhat  too-high  estimates  for 
larger  values  of  the  simultaneous  overflow  calls  n. 

Fig.  16  shows  additional  comparisons  of  overflow  and  negative  bi- 
nomial distributions.  As  before,  the  agreement  is  quite  satisfactory  to 
0.01  probability,  the  negative  binomial  thereafter  tending  to  give  some- 
what high  values. 

On  Fig.  17  are  compared  the  individual  6(n)  density  distributions  for 
several  cases.  The  agreement  of  the  negative  binomial  with  the  true 
distribution  is  seen  to  be  uniformly  good.  The  dots  indicate  the  random 
(Poisson)  individual  term  distribution  corresponding  to  the  a  =  9.6  case- 


1.0 

"T*^ 

;J-^ 

— 

TRUE    DISTRIBUTION 

\ 

^^^^^\- 

_ 



NEGATIVE    BINOMIAL 

\ 

<^ 

• 

\ 

FITTING    DISTRIBUTION 

CORRESPONDING 
RANDOM  TRAFFIC 

0.1 

-\ 

\ 

> 

v 

\ 

_     \ 

•    \ 

•\ 

\ 

n) 

\ 

\     ^ 

\ 

»  \\     \ 

\                     • 

\ 

^  V  \ 

0.01 

- 

\ 

V5 

•  v\ 

\s:=io 

\\ 

\\   \ 

. 

^ 

V              \>    V 

•       \ 

• 

\  ^^     \  n^ 

0.001 

_J M       \   i           1 \   l>    \V 1 1 

0       t       2      3      4       5      6      7      8       9      10     11      12     13     14     15 
n  =  NUMBER    OF    SIMULTANEOUS    CALLS 

Fig.  14  —  Probability  distributions  of  overflow  traffic  with  5  erlangs  offered  to 
1,  2,  5,  and  10  trunks,  fitted  by  negative  binomial. 


I 


THEORIES   FOR   TOLL   TRAFFIC   ENGINEERING   IN   THE   U.   S.   A.      457 

the  agreement,  of  course,  is  poor  since  the  non-randomness  of  the  over- 
flow here  is  marked,  having  an  average  of  1.88  and  a  variance  of  3.84. 

Comparison  of  Negative  Binomial  with  Overflow  Distributions  Observed 
hi/  llirowdoivns  and  on  Actual  Trunk  Groups 

Fig.  18  shows  a  comparison  of  the  negative  binomial  with  the  over- 
How  distributions  from  four  direct  groups  as  seen  in  throwdown  studies, 
'ilie  agreement  over  the  range  of  group  sizes  from  one  to  fifteen  trunks  is 
seen  to  be  excellent.  The  assumption  of  randomness  (Poisson)  as  shown 
by  the  dot  values  is  clearly  unsatisfactory  for  overflows  beyond  more 
than  two  or  three  trunks. 

A  number  of  switch  counts  made  on  the  final  group  of  an  operating 
toll  alternate  routing  system  at  Newark,  New  Jersey,  during  periods 
when  few  calls  were  lost,  have  also  shown  good  agreement  with  the  neg- 
ative binomial  distribution. 

7.2.2.  A  Probability  Distribution  for  Combined  Overflow  Traffic  Loads 

It  has  been  shown  in  Section  7.2.1  that,  at  least  for  load  ranges  of  wide 
interest,  the  negative  binomial  with  but  two  parameters,  chosen  to  agree 


Fx(§n) 


0.01 


0.001 


TION 

OMIAL 
BUTION 


0       I       2      3      4       5      6      7       8      9      10      11     12     13     14     15 
n=  NUMBER   OF   SIMULTANEOUS   CALLS 


Fig.  15  —  Probability  distributions  of  overflow  traffic  with  5  erlangs  offered  to 
1,  2,  5,  and  10  trunivs,  when  all  trunks  are  busy;  fitted  by  negative  binomial. 


458 


THE   BELL   SYSTEM   TECHNICAL   JOURNAL,   MARCH    1956 


with  mean  and  variance,  gives  a  satisfactory  jfit  to  the  distribution  of 
traffic  overflowing  a  group  of  trunks.  It  is  now  possible,  of  course,  to 
convohite  the  various  overflows  from  any  number  of  groups  of  varying 
sizes,  to  obtain  a  combined  overflow  distribution.  This  procedure,  how- 
ever, would  be  very  clumsy  and  laborious  since  at  each  switching  point 
in  the  toll  alternate  route  system  an  entirely  difl"erent  layout  of  loads  and 
high  usage  groups  would  require  solution;  it  would  be  unfeasible  for 
practical  working. 

We  return  again  to  the  method  of  moments.  Since  the  overflows  of 
the  several  high  usage  groups  will,  in  general,  be  independent  of  one 
another,  the  iih  semi-invariants  Xi  of  the  individual  overflows  can  be 
combined  to  give  the  corresponding  semi-invariants  A,  of  their  total, 


Ai  —  iXi  +  2X1  + 


(27) 


Or,  in  terms  of  the  overflow  means  and  variances,  the  corresponding 
parameters  of  the  combined  loads  are 

Average  =  A'  =  ai  -{-  az  +  ■  ■  ■  (28) 

Variance  =  V  =  vi  +  V2  +  •  •  -  (29) 


TRUE    DISTRIBUTION 


NEGATIVE    BINOMIAL 

FITTING   DISTRIBUTION 


0.001 


2      3      4       5      6       7       8      9      10     II      12     13     14    15 
n  =  NUMBER    OF  SIMULTANEOUS    CALLS 


Fig.  16  —  Probability  distributions  of  overflow  traffic:  3  erlangs  offered  to  2 
trunks,  and  9.6  erlangs  offered  to  10  trunks. 


THEORIES   FOR   TOLL   TRAFFIC   ENGINEERING   IN   THE   U.    S.    A.      459 

With  the  mean  and  variance  of  the  combined  overflows  now  deter- 
mined, the  negative  binomial  can  again  be  employed  to  give  an  approxi- 
mate description  of  the  distribution  of  the  simultaneous  calls  (p{z)  offered 
to  the  common,  or  alternate,  group. 

The  acceptability  of  this  procedure  can  be  tested  in  various  ways.  One 
way  is  to  examine  whether  the  convolution  of  several  negative  binomials 
(representing  overflows  from  individual  groups)  is  sufficiently  well  fitted 
by  another  negative  binomial  with  appropriate  mean  and  variance,  as 
found  above. 

It  can  easily  be  shown  that  the  convolution  of  several  negative  bi- 
nomials all  with  the  same  over-dispersion  (variance-to-mean  ratio)  but 
not  necessarily  the  same  mean,  is  again  a  negative  binomial.  Shown  in 
Table  I  are  the  distribution  components  and  their  parameters  of  two 
examples  in  which  the  over-dispersion  parameters  are  not  identical.  The 
third  and  fourth  semi-invariants  of  the  fitted  and  fitting  distributions,  are 
seen  to  diverge  considerably,  as  do  the  Pearsonian  skewness  and  kurtosis 
factors.  The  test  of  acceptability  for  traffic  fluctuation  description  comes 
in  comparing  the  fitted  and  fitting  distributions  which  are  shown  on 
Fig.  19.  Here  it  is  seen  that,  despite  what  might  appear  alarming  dis- 


0(n) 


0.01 


O.OOI 


TRUE    DISTRIBUTION 


NEGATIVE    BINOMIAL 

FITTING   DISTRIBUTION 

•    RANDOM  TRAFFIC,    8=1.9 


a  =  9.6 

=  3.84 


I         2        3        4        5        6       7        8        9       10       II       12 

n  =  NUMBER    OF    SIMULTANEOUS    CALLS 


Fig.  17  —  Probability  density  distributions  of  overflow  traffic  from  10  trunks, 
fitted  by  negative  binomial. 


460 


THE   BELL   SYSTEM   TECHNICAL   JOUENAL,    MARCH    1956 


parities  in  the  higher  semi-invariants,  the  agreement  for  practical  traffic 
purposes  is  very  good  indeed. 

Numerous  throwdown  checks  confirm  that  the  negative  binomial  em- 
ploying the  calculated  sum-overflow  mean  and  variance  has  a  wide  range 
over  which  the  fit  is  quite  satisfactory  for  traffic  description  purposes. 
Fig.  20  shows  three  such  trunking  arrangements  selected  from  a  con- 
siderable number  which  have  been  studied  by  the  simulation  method. 
Approximate!}^  5,000,  3,500,  and  580  calls  were  run  through  in  the  three 
examples,  respective!}' .  Tlie  overflow  parameters  obtained  !)y  experiment 
are  seen  to  agree  reasonably  well  with  the  theoretical  ones  from  (28) 
and  (29)  when  the  numbers  of  calls  processed  is  considered. 

On  Fig.  21  are  sliown,  for  the  first  arrangement  of  Fig.  20,  distributions 
of  simultaneous  offered  calls  in  each  subgroup  of  trunks  compared  with 
the  corresponding  Poisson;  the  agreement  is  satisfactory  as  was  to  be 
expected.  The  sum  distribution  of  the  overflows  from  the  eight  subgroups 
is  given  at  the  foot  of  the  figure.  The  superposed  Poisson,  of  course,  is  a 
poor  fit;  the  negative  binomial,  on  the  other  hand,  appears  quite  accept- 
able as  a  fitting  curve. 


1.0 
0.8 
0.6 


P  2n 


1   TRUNK-  a  =  \.22 


3  TRUNKS-  a  =  2.24 


0.4  - 


0.2  ■ 


1.0 


0.8  - 


0.6 
0.4 
0.2 


234501  234 

n=NUMBER   OF  SIMULTANEOUS  CALLS 


THEORY 

OBSD 

V\ 

( ) 

( ) 

AVG          0.67 

0.63 

VAR          0.77 

0.60 

i       •    RANDOM  TRAFFIC 

\,                     a  =  0.67 

THEORY        OBSD 

c- )    ( — 1 

u 

AVG         0.55 

0.51 

VAR          0.77 

0.63 

\\ 

•    RANDOM 

TRAFFIC 

a= 

D.55 

v^^ 

P^n 


1.0 

15  TRUNKS-  a 
\                          THEORY 

=  11.46 
OBSD      '-O 

.\                         ( H 

( ) 

0.8 

*\              AVG          0.81 

0.80       '-'•® 

'A            VAR          1.88 

1.42 

0.6 

"\\,            •  RANDOM  TRAFFIC      °-^ 

\l                   a=o.8i 

0.4 

0.4 

0.2 

0.2 

0 

•  ^'^v,.^^^^ 

_      ,       n 

9  TRUNKS-  a  =  6.21 

THEORY       OBSD 
( -)       ( ) 

AVG  0.52  0.46 

VAR  1.00  1.48 

.    RANDOM  TRAFFIC 
a  =  0.52 


4  68  10  024  68 

n=NUMBER   OF  SIMULTANEOUS  CALLS 


10 


12 


Fifj;.  18  —  Ovorflow  (li.-<t ril)utioiis  from  diroct  interoffice  trunk  groups;  negative 
binomial  theory  versus  thrgwclowji  observations. 


THEORIES   FOR  TOLL  TRAFFIC   ENGINEERING  IN  THE   U.   S.   A.      461 

Table  I  —  Comparison  of  Parameters  of  a  Fitting 

Negative  Binomial  to  the  Convolution  of 

Three  Negative  Binomials 


Example  No.  1 

Example  No.  2 

Component 

Component 

parameters 

Component 
dist'n  No. 

Component  parameters 

dist'n  No. 

Mean 

Variance 

Mean 

Variance 

1 

5 

5 

1 

1 

1 

2 

2 

4 

2 

2 

3 

3 

1 

3 

3 

2 

6 

8 

12 

5 

10 

Semi-Invariants  A,  Skewness  \/pi  ,  and  Kurtosis  ^2  ,  of  Sum  Distributions 


Parameter 

E.xact 

Fitting 

Parameter 

Exact 

Fitting 

Ai 

8 

8 

Ai 

5 

5 

Ao 

12 

12 

A2 

10 

10 

As 

32 

24 

As 

37 

30 

A4 

168 

66 

A4 

239.5 

130 

VFi 

0.770 

0.577 

V/3i 

1.170 

0.949 

/32 

4.167 

3.458 

/32 

5.395 

4.300 

Fig.  22  shows  the  corresponding  comparisons  of  the  overflow  loads  in 
the  other  two  trunk  arrangements  of  Fig.  20.  Again  good  agreement 
with  the  negative  binomial  is  seen. 


7.3.  Equivalent  Random  Theory  for  Prediction  of  Amount  of  Traffic  Over- 
flowing a  Single  Stage  Alternate  Route,  and  Its  Character,  with  Lost 
Calls  Cleared 

As  discussed  in  Section  7.2,  when  random  traffic  is  offered  to  a  limited 
number  of  trunks  x,  the  overflow  traffic  is  well  described  (at  least  for 
traffic  engineering  purposes)  by  the  two  parameters,  mean  a  and  variance 
V.  The  result  can  readily  be  applied  to  a  group  divided  (in  one's  mind) 
two  or  more  times  as  in  Fig.  23. 

Employing  the  a  and  v  curves  of  Figs.  12  and  13,  and  the  appropriate 
numbers  of  trunks  a;i  ,  Xi  +  0:2 ,  and  Xi  +  X2  +  x^ ,  the  pairs  of  descrip- 
tive parameters,  ai  ,  vi  ,  ao ,  vo  and  a-s ,  v-a  can  be  read  at  once.  It  is  clear 
then  that  if  at  some  point  in  a  straight  multiple  a  traffic  with  parameters 
ai  ,  Vi  is  seen,  and  it  is  offered  to  .r2  paths,  the  overflow  therefrom  will 
have  the  characteristics  012 ,  vo  .  To  estimate  the  particular  values  of  a-y 
and  v-i ,  one  would  first  determine  the  values  of  the  equivalent  random 


462 


THE    BELL   SYSTEM   TECHNICAL   JOURNAL,    MARCH    1956 


P5n 


P^n 


CONVOLUTION  OF  3  NEGATIVE    BINOMIAL 
VARIABLES    WITH    PARAMETERS: 

AVG     WR 

1  I 

2  3 
2  6 

, -FITTING  NEGATIVE    BINOMIAL 


6  8  10  12 

n=  NUMBER  OF   CALLS    PRESENT 


I 


-I I l_^ 


14 


16 


Fig.  19  —  Fitting  sums  of  negative  binomial  variables  with  a  negative  binomial. 


traffic  a  and  trunks  .Ti  which  would  have  produced  ai  and  Vi  .  Then  pro- 
ceeding in  the  forward  direction,  using  a  and  Xi  +  X2 ,  one  consults  the 
a  and  v  charts  to  find  txi  and  Vz .  Thus,  within  the  limitations  of  straight 
group  traffic  flow,  the  character  (mean  and  variance)  of  any  overflow 
load  from  x  trunks  can  be  predicted  if  the  character  (mean  and  variance) 
of  the  load  submitted  to  them  is  known. 

Curves  could  be  constructed  in  the  manner  just  described  by  which  the 
overflow's  a'  and  v'  are  estimated  from  a  load,  a  and  v,  offered  to  x  trunks. 
An  illustrative  fragment  of  such  curves  is  shown  in  Appendix  II,  with  an 
example  of  their  application  in  the  calculation  of  a  straight  trunk  group 
loss  by  considering  the  successive  overflows  from  each  trunk  as  the 
offered  loads  to  the  next. 

Enough,  perhaps,  has  been  shown  in  Section  7.2  of  the  generally  ex- 
cellent descriptions  of  a  variety  of  non-random  traffic  loads  obtainable 
by  the  use  of  only  the  two  parameters  a  and  v,  to  make  one  strongly 
suspect  that  most  of  the  fluctuation  information  needed  for  traffic  engi- 
neering purposes  is  contained  in  those  two  values.  If  this  is,  in  fact,  the 
case,  we  should  then  be  able  to  predict  the  overflow  a',  v'  from  x  trunks 


THEORIES    FOR   TOLL  TRAFFIC   ENGINEERING   IN   THE   U.    S.    A.      463 

\\ith  an  offered  load  a,  v  which  has  arisen  in  any  manner  of  overflow  from 
earlier  high  usage  groups,  as  illustrated  in  Fig.  24. 

This  is  found  to  be  the  case,  as  will  be  illustrated  in  several  studies  de- 
scribed in  the  balance  of  this  section.  In  the  determination  of  the  charac- 
teristics of  the  overflow  traffic  a',  v'  in  the  cases  of  non-full-access  groups, 
such  as  Figs.  24(b)  and  24(c),  the  equivalent  straight  group  is  visualized 
[Fig.  24(a)],  and  the  Eciuivalent  Random  load  A  and  trunks  S  are  found.* 
I  Using  A,  and  *S  +  C,  to  enter  the  a  and  v  curves  of  Figs.  12  and  13,  a 
,  and  v'  are  readily  determined.  To  facilitate  the  reading  of  .1  and  S,  Fig. 
25 1  and  Fig.  26 f  (which  latter  enlarges  the  lower  left  corner  of  Fig.  25) 
have  been  drawn.  Since,  in  general,  a  and  v  will  not  have  come  from  a 
simple  straight  group,  as  in  Fig.  24(a),  it  is  not